critical f(x)=6t^{2/3}+t^{5/3}

{ "query": { "display": "critical points $$f\\left(x\\right)=6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}$$", "symbolab_question": "FUNCTION#critical f(x)=6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "critical", "default": "t=-\\frac{12}{5},t=0" }, "steps": { "type": "interim", "title": "Critical Points of $$6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}:{\\quad}t=-\\frac{12}{5},\\:t=0$$", "input": "6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}", "steps": [ { "type": "definition", "title": "Critical point definition", "text": "Critical points are points where the function is defined and its derivative is zero or undefined" }, { "type": "interim", "title": "Find where $$f^{\\prime}\\left(t\\right)$$ is equal to zero or undefined", "input": "6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}", "result": "t=-\\frac{12}{5},\\:t=0", "steps": [ { "type": "interim", "title": "$$f^{\\prime}\\left(t\\right)=\\frac{12+5t}{3t^{\\frac{1}{3}}}$$", "input": "\\frac{d}{dt}\\left(6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dt}\\left(6t^{\\frac{2}{3}}\\right)+\\frac{d}{dt}\\left(t^{\\frac{5}{3}}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(6t^{\\frac{2}{3}}\\right)=\\frac{4}{t^{\\frac{1}{3}}}$$", "input": "\\frac{d}{dt}\\left(6t^{\\frac{2}{3}}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dt}\\left(t^{\\frac{2}{3}}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6\\cdot\\:\\frac{2}{3}t^{\\frac{2}{3}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$6\\cdot\\:\\frac{2}{3}t^{\\frac{2}{3}-1}:{\\quad}\\frac{4}{t^{\\frac{1}{3}}}$$", "input": "6\\cdot\\:\\frac{2}{3}t^{\\frac{2}{3}-1}", "result": "=\\frac{4}{t^{\\frac{1}{3}}}", "steps": [ { "type": "interim", "title": "$$t^{\\frac{2}{3}-1}=t^{-\\frac{1}{3}}$$", "input": "t^{\\frac{2}{3}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{2}{3}-1:{\\quad}-\\frac{1}{3}$$", "input": "\\frac{2}{3}-1", "result": "=t^{-\\frac{1}{3}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=-\\frac{1\\cdot\\:3}{3}+\\frac{2}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:3+2}{3}" }, { "type": "interim", "title": "$$-1\\cdot\\:3+2=-1$$", "input": "-1\\cdot\\:3+2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=-3+2" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-3+2=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SvYRo5M5eaqKA092Lzw/V1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBrscnvl1118p3R4GZ3W81EVg=" } }, { "type": "step", "result": "=\\frac{-1}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7T6cHmV99UoyF+Un4NIPAQu0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjcATzyvwdKDRDlzzd0Kla5wH2kDe5DGYTz3TrPquGdIhZoeQg27gSlcodSF6rKR4IGuRLSj9jrA8t4VRzhiUQpREDGbsuNaklPQr2a3AJvF8=" } }, { "type": "step", "result": "=6\\cdot\\:\\frac{2}{3}t^{-\\frac{1}{3}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$t^{-\\frac{1}{3}}=\\frac{1}{t^{\\frac{1}{3}}}$$" ], "result": "=6\\cdot\\:\\frac{2}{3}\\cdot\\:\\frac{1}{t^{\\frac{1}{3}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{2\\cdot\\:1\\cdot\\:6}{3t^{\\frac{1}{3}}}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1\\cdot\\:6=12$$", "result": "=\\frac{12}{3t^{\\frac{1}{3}}}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{3}=4$$", "result": "=\\frac{4}{t^{\\frac{1}{3}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wHsyNILr7YqAxQ7kxdsiXvVPSPPxvb7XIuW4IuHeRzNzp5U103zCpi/Gm21jPxn4dYPfXQvX4/bINBB8wSEQ0YvHIZnpqiK63vE+Pcs0CegaL1UpkG2ugXfqT4cauyHHP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXT0gainIBHCDBcPnB9nWFHt9iiZczbb2b1rf1DaPLifdquEfcPm/XueXywwkTXEPKI=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(t^{\\frac{5}{3}}\\right)=\\frac{5t^{\\frac{2}{3}}}{3}$$", "input": "\\frac{d}{dt}\\left(t^{\\frac{5}{3}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{5}{3}t^{\\frac{5}{3}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{5}{3}t^{\\frac{5}{3}-1}:{\\quad}\\frac{5t^{\\frac{2}{3}}}{3}$$", "input": "\\frac{5}{3}t^{\\frac{5}{3}-1}", "result": "=\\frac{5t^{\\frac{2}{3}}}{3}", "steps": [ { "type": "interim", "title": "$$t^{\\frac{5}{3}-1}=t^{\\frac{2}{3}}$$", "input": "t^{\\frac{5}{3}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{5}{3}-1:{\\quad}\\frac{2}{3}$$", "input": "\\frac{5}{3}-1", "result": "=t^{\\frac{2}{3}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=-\\frac{1\\cdot\\:3}{3}+\\frac{5}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:3+5}{3}" }, { "type": "interim", "title": "$$-1\\cdot\\:3+5=2$$", "input": "-1\\cdot\\:3+5", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=-3+5" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-3+5=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NKzIdLoxfOWctINKoYkNv1XTSum/z5kLpMzXS1UJIezBOIcT9LwoPFpLgX7k/D9TSuvnmHHmwzaA2REu2aAnP84ka8F9WsyASLd6olVazQQ=" } }, { "type": "step", "result": "=\\frac{2}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bJ+kkIGi5/ZQrlXHVkwB3u0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjlzMlYuD/GiUtiBG+t9ydqv8//6/nV5O4fb8Xgwi7mapeixdzicS1m80h4uhRMenKyak9KT7njyg2Gkgetrdeds67uJicgMpLzW5Yu+HQ/jQ=" } }, { "type": "step", "result": "=\\frac{5}{3}t^{\\frac{2}{3}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{5t^{\\frac{2}{3}}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qN5CnGdRfJL68RhZyKjjNLCttvJlYc8bs4y+5RPcRqgAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJi2YcffPCbT3NLMHZYzoJbHDakV5xtzrC6sUYTlsYKLn1sD7NfhsPe7eDHrmjY0mE8DwQcN28geUT7aUsumC7drTLEo3PiRqst3iDDSngBcH2sn2RC+c/BDkqFyxK4HceQ==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{4}{t^{\\frac{1}{3}}}+\\frac{5t^{\\frac{2}{3}}}{3}" }, { "type": "interim", "title": "Simplify $$\\frac{4}{t^{\\frac{1}{3}}}+\\frac{5t^{\\frac{2}{3}}}{3}:{\\quad}\\frac{12+5t}{3t^{\\frac{1}{3}}}$$", "input": "\\frac{4}{t^{\\frac{1}{3}}}+\\frac{5t^{\\frac{2}{3}}}{3}", "result": "=\\frac{12+5t}{3t^{\\frac{1}{3}}}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$t^{\\frac{1}{3}},\\:3:{\\quad}3t^{\\frac{1}{3}}$$", "input": "t^{\\frac{1}{3}},\\:3", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$t^{\\frac{1}{3}}$$ or $$3$$", "result": "=3t^{\\frac{1}{3}}" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$3t^{\\frac{1}{3}}$$" }, { "type": "step", "primary": "For $$\\frac{4}{t^{\\frac{1}{3}}}:\\:$$multiply the denominator and numerator by $$3$$", "result": "\\frac{4}{t^{\\frac{1}{3}}}=\\frac{4\\cdot\\:3}{t^{\\frac{1}{3}}\\cdot\\:3}=\\frac{12}{3t^{\\frac{1}{3}}}" }, { "type": "step", "primary": "For $$\\frac{5t^{\\frac{2}{3}}}{3}:\\:$$multiply the denominator and numerator by $$t^{\\frac{1}{3}}$$", "result": "\\frac{5t^{\\frac{2}{3}}}{3}=\\frac{5t^{\\frac{2}{3}}t^{\\frac{1}{3}}}{3t^{\\frac{1}{3}}}=\\frac{5t^{\\frac{2}{3}+\\frac{1}{3}}}{3t^{\\frac{1}{3}}}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{12}{3t^{\\frac{1}{3}}}+\\frac{5t^{\\frac{2}{3}+\\frac{1}{3}}}{3t^{\\frac{1}{3}}}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{12+5t^{\\frac{2}{3}+\\frac{1}{3}}}{3t^{\\frac{1}{3}}}" }, { "type": "interim", "title": "Combine the fractions $$\\frac{2}{3}+\\frac{1}{3}:{\\quad}1$$", "result": "=\\frac{12+5t^{1}}{3t^{\\frac{1}{3}}}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{2+1}{3}" }, { "type": "step", "primary": "Refine", "result": "=1" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$a^{1}=a$$", "secondary": [ "$$t^{1}=t$$" ], "result": "=\\frac{12+5t}{3t^{\\frac{1}{3}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XzeR5fQtb5Te692pE/GE8ikPrBmof/cPtaYGidjGHkWWXF1hs8aDg3VQmCAfnvsbbxiOACY2YlPhqzR+zvdPFc0ag8T1MwTer44+aCS/ZFB/jbWPuyH1f+vj8lAyZN4mF5zecSUDv1a3IYHzXCNra8ahA2VuXun+dokKd6eMdc1N5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+eCz/YkutMkYWTlpzWqJmi+x7Fl0RqNAiPtgS30wOGj3IUY/HMXN3JEy4nbeHIo6UVdl41TRHpIfCbRt2Vw5U0E=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Solve $$\\frac{12+5t}{3t^{\\frac{1}{3}}}=0:{\\quad}t=-\\frac{12}{5}$$", "input": "\\frac{12+5t}{3t^{\\frac{1}{3}}}=0", "steps": [ { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "12+5t=0" }, { "type": "interim", "title": "Solve $$12+5t=0:{\\quad}t=-\\frac{12}{5}$$", "input": "12+5t=0", "steps": [ { "type": "interim", "title": "Move $$12\\:$$to the right side", "input": "12+5t=0", "result": "5t=-12", "steps": [ { "type": "step", "primary": "Subtract $$12$$ from both sides", "result": "12+5t-12=0-12" }, { "type": "step", "primary": "Simplify", "result": "5t=-12" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5t=-12", "result": "t=-\\frac{12}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5t}{5}=\\frac{-12}{5}" }, { "type": "step", "primary": "Simplify", "result": "t=-\\frac{12}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "t=-\\frac{12}{5}" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}t=-\\frac{12}{5}\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\frac{12+5t}{3t^{\\frac{1}{3}}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$t=-\\frac{12}{5}:{\\quad}$$True", "input": "\\frac{12+5\\left(-\\frac{12}{5}\\right)}{3\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}}=0", "steps": [ { "type": "interim", "title": "$$\\frac{12+5\\left(-\\frac{12}{5}\\right)}{3\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}}=0$$", "input": "\\frac{12+5\\left(-\\frac{12}{5}\\right)}{3\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{12-5\\cdot\\:\\frac{12}{5}}{3\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}}" }, { "type": "interim", "title": "$$3\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}=-3\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}$$", "input": "3\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}", "steps": [ { "type": "interim", "title": "$$\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}=-\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}$$", "input": "\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{\\frac{1}{n}}=-\\left(a\\right)^{\\frac{1}{n}},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-\\frac{12}{5}\\right)^{\\frac{1}{3}}=-\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}$$" ], "result": "=-\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AW0fUdECBwMOtE19brxdifHl/kt0mc6Jh9jPSXCm4ER8kR7hsO/rTOTBE0w4+r1Re5K9X3PealwsoGbmbO7jPn7gLfkzaTAQCPXT6ISF3Eqnz8Qe7zC/DbVy2YoV0kZPV9VLPTDRDo74oHfifpLRiVMBZY9s6r5tWhTAMXzk7Ih9wGfBh+hIXnDpxU0vJ4unUwFlj2zqvm1aFMAxfOTsiMxfla5SPZR4jgQ6n25d37k=" } }, { "type": "step", "result": "=3\\left(-\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-3\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73Vg4/tpPPTx+4GpEGk2nuu5h4xdYtubViDWdKxHxVstFdiKL0bo+0lLRUzNXm1IEebTKSUwQXHFFpEsrzHtUqXiX27+5czz2MT6ns7HJh4hRMtwUzUKzFth5oUmvA1l5ZgbKfKW0HJe3lrhv5BsQrWQ/l5LJR1p8Z/dWm0tVDSsE9g5LyzKX3sVO581JNPSNdz7YMZJ8K+P0zkLlq5ffuVOAZt/oqkgX1h2ckcEMQKqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{12-5\\cdot\\:\\frac{12}{5}}{-3\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{12-5\\cdot\\:\\frac{12}{5}}{3\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}}" }, { "type": "interim", "title": "$$\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}=\\frac{12^{\\frac{1}{3}}}{5^{\\frac{1}{3}}}$$", "input": "\\left(\\frac{12}{5}\\right)^{\\frac{1}{3}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{12^{\\frac{1}{3}}}{5^{\\frac{1}{3}}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70FBf/fQd22I+9SY3Y5+LflOAZt/oqkgX1h2ckcEMQKoAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJgOQ1Wbxyb83OGx0M6YgANBR0tCqC10v9lOEQ/TQPstUaAp/PsrLe3sDKAsTOfuqLL0lR0uL2U8Uh/bwQUYU8KpkAv4W4enzINZX6io2Z3tCjkoySx7AUd0sURsgFCSxWBovVSmQba6Bd+pPhxq7Icd2yV+a6eGOeeztTJfvg/OvFkVDw5yEF2DEw4msQzKTew==" } }, { "type": "step", "result": "=-\\frac{-5\\cdot\\:\\frac{12}{5}+12}{3\\cdot\\:\\frac{12^{\\frac{1}{3}}}{5^{\\frac{1}{3}}}}" }, { "type": "interim", "title": "$$5\\cdot\\:\\frac{12}{5}=12$$", "input": "5\\cdot\\:\\frac{12}{5}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{12\\cdot\\:5}{5}" }, { "type": "step", "primary": "Cancel the common factor: $$5$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mAcCRZLNgqmjEVD/La84+4dShV1JTBo595blmlNM5bkJQJZuTAY5js+oqjdT8kslOSPHu1G49xWGY3ncXeazfn2stX9ZrjmG8JBkmRp0rPNocR5KdxhXZnxd5zoBInq+JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=-\\frac{12-12}{3\\cdot\\:\\frac{12^{\\frac{1}{3}}}{5^{\\frac{1}{3}}}}" }, { "type": "step", "primary": "Subtract the numbers: $$12-12=0$$", "result": "=-\\frac{0}{3\\cdot\\:\\frac{12^{\\frac{1}{3}}}{5^{\\frac{1}{3}}}}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YSIQzMheLvEWWwrgYp7nukQMr0PeP4fxnJVuHPrgvOhkP5eSyUdafGf3VptLVQ0rGi9VKZBtroF36k+HGrshxwCWKUbvV6WK3fDUgFtg3Q8GS7g7rdYb4CeWQePA28lVo3oe/oyhMy2+1TQhDBd2f++g4CBUxRCYzSozB7vhthYNiVIPPCgYVlgyaG3OdVdGqIqm5aPSe0dCrvwT6ct3nQLW6St+Q7wa7tYB9e6A4z8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "t=-\\frac{12}{5}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Find undefined (singularity) points of $$f^{\\prime}\\left(t\\right):{\\quad}t=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{12+5t}{3t^{\\frac{1}{3}}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$3t^{\\frac{1}{3}}=0:{\\quad}t=0$$", "input": "3t^{\\frac{1}{3}}=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3t^{\\frac{1}{3}}=0", "result": "t^{\\frac{1}{3}}=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3t^{\\frac{1}{3}}}{3}=\\frac{0}{3}" }, { "type": "step", "primary": "Simplify", "result": "t^{\\frac{1}{3}}=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Take both sides of the equation to the power of $$3:{\\quad}t=0$$", "input": "t^{\\frac{1}{3}}=0", "result": "t=0", "steps": [ { "type": "step", "result": "\\left(t^{\\frac{1}{3}}\\right)^{3}=0^{3}" }, { "type": "interim", "title": "Expand $$\\left(t^{\\frac{1}{3}}\\right)^{3}:{\\quad}t$$", "input": "\\left(t^{\\frac{1}{3}}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=t^{\\frac{1}{3}\\cdot\\:3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{3}\\cdot\\:3=1$$", "input": "\\frac{1}{3}\\cdot\\:3", "result": "=t", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71Hcipo16ybrRdbQSFyXFLyPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6ndrE5pRJHjGLuD4z7ETvZP4l/Cp3Oq6hZvUc7OAI8rhp" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e4b5T0h7TIdFeT6H4/gVBRt6mG8EUr5xjkEMVXgVUCMLAZlDhoAdFQF6AF4pPagvRKAKbro8EavzLN8GuIDU/PWVUjtAdd6EqSvsyX6r+5AezFilETfuNygjs0XPkV2h2ZxBEJOL2iu4GK90L3LW5frwiqMcGAvMzJY8YdUQdyY=" } }, { "type": "interim", "title": "Expand $$0^{3}:{\\quad}0$$", "input": "0^{3}", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7INho7VOwyV4G1diIPM8OSw7afGBEifydK7wCwslogi9F5bLdlaCFshD3v4rLy7EcFU0qZ4XMwhSlm3knFkjDUN7UD16RqFQha281as3i/3j5R4T0LNXcFLIYB+gGS/if" } }, { "type": "step", "result": "t=0" } ], "meta": { "interimType": "Take Both Sides To Power Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjCnr0+SqZ1B7Ftmqgi8CPcUZDwlGiBvBzElPqp/PNkG8uuQTC2rJk2wnzJ36NyzAtVt1yuef+bXpfRUhDTkdaSM56CttsxUJUXr89NKaLFzBR1JB/LpcuDrHITnyYRg7vK4UbfLSkcUuA0BIMx/V2rz1mtdYTQ2zWbr8Ksm8OspQZfk5JnQTZE7/ywIhP0W4+RC0BDAQEwNVzc0oe74szxMJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Verify Solutions:$${\\quad}t=0\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$3t^{\\frac{1}{3}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$t=0:{\\quad}$$True", "input": "3\\cdot\\:0^{\\frac{1}{3}}=0", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:0^{\\frac{1}{3}}=0$$", "input": "3\\cdot\\:0^{\\frac{1}{3}}", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{\\frac{1}{3}}=0$$" ], "result": "=3\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sLGvCGeS3hfD/Xw+DQI36b0GkbSrtjQ15A7/bcJn9rTNGoPE9TME3q+OPmgkv2RQz7FeFrf3rcM8IZlDz2c0drrgkyCMPB58MhCQWH3BBXQJbzLDfwKck1AG3VapOMm9EEWJXNA8Rj59tQEICFmFoA==" } }, { "type": "step", "primary": "$$0=0$$" }, { "type": "step", "result": "\\mathrm{True}" } ], "meta": { "interimType": "Generic Plug 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "t=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "t=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "result": "t=-\\frac{12}{5},\\:t=0" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/xxoZoVpsLc4CEilkpPaZjLoBIfH6rTsh0/l5JgCipMjsVpIIcv9JNADXPMrhTVjbtCFNukBOT9dsM4NfKhvf/eahXPsVJTOjQvbWyGFjY+Y+PFVMLZiIUzbm52o53IfJ+fz69t01/AJqj26Ph79PnpPvbBmbuQNTF0TphKZ8RuvaJ60y62t83JPjOFt4iA7YZFBIvGuEiPGnKvBdQV5qHvBEjKOtooLDm7sJWUJLcPENvG7cGF7u0Zxdm3zFEMqNA5alFN/7eYB3esHPT9qlUdc=" } }, { "type": "step", "primary": "Identify critical points not in $$f\\left(t\\right)$$ domain" }, { "type": "interim", "title": "Domain of $$6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}\\::{\\quad}-\\infty\\:<t<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<t<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "step", "primary": "All critical points are in domain", "result": "t=-\\frac{12}{5},\\:t=0" } ], "meta": { "solvingClass": "Function Critical" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "6t^{\\frac{2}{3}}+t^{\\frac{5}{3}}" }, "showViewLarger": true } } }