critical f(x)=(ln(x))/(x^6)

{ "query": { "display": "critical points $$f\\left(x\\right)=\\frac{\\ln\\left(x\\right)}{x^{6}}$$", "symbolab_question": "FUNCTION#critical f(x)=\\frac{\\ln(x)}{x^{6}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "critical", "default": "x=\\sqrt[6]{e}" }, "steps": { "type": "interim", "title": "Critical Points of $$\\frac{\\ln\\left(x\\right)}{x^{6}}:{\\quad}x=\\sqrt[6]{e}$$", "input": "\\frac{\\ln\\left(x\\right)}{x^{6}}", "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 $$f^{\\prime}\\left(x\\right)$$ is equal to zero or undefined", "input": "\\frac{\\ln\\left(x\\right)}{x^{6}}", "result": "x=\\sqrt[6]{e},\\:x=0", "steps": [ { "type": "interim", "title": "$$f^{\\prime}\\left(x\\right)=\\frac{1-6\\ln\\left(x\\right)}{x^{7}}$$", "input": "\\frac{d}{dx}\\left(\\frac{\\ln\\left(x\\right)}{x^{6}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)x^{6}-\\frac{d}{dx}\\left(x^{6}\\right)\\ln\\left(x\\right)}{\\left(x^{6}\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "result": "=\\frac{1}{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhHxrkiFdmQgNsZN21633mEcjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJlc0OBMs8qTL4oWnxx62vyRTW26qciuyUBGXQExCUedYi3kiAkvXOTkrmcfV8WHLnF4CmnHjYZyazvJkuCAZs64=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{6}\\right)=6x^{5}$$", "input": "\\frac{d}{dx}\\left(x^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6x^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6x^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYu5iW3FGajTt+lS6EjZ9mTak3hxk9aCfAWodBRxXgUexpGh0eWPglcqaUbPRp1wW2f8//6/nV5O4fb8Xgwi7maqO95Is7YBdcRqXmTH8euc2rweoD+FNNYz/CCNtWBDj5A==" } }, { "type": "step", "result": "=\\frac{\\frac{1}{x}x^{6}-6x^{5}\\ln\\left(x\\right)}{\\left(x^{6}\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{x}x^{6}-6x^{5}\\ln\\left(x\\right)}{\\left(x^{6}\\right)^{2}}:{\\quad}\\frac{1-6\\ln\\left(x\\right)}{x^{7}}$$", "input": "\\frac{\\frac{1}{x}x^{6}-6x^{5}\\ln\\left(x\\right)}{\\left(x^{6}\\right)^{2}}", "result": "=\\frac{1-6\\ln\\left(x\\right)}{x^{7}}", "steps": [ { "type": "interim", "title": "$$\\left(x^{6}\\right)^{2}=x^{12}$$", "input": "\\left(x^{6}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=x^{6\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=x^{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ECzgnbX99BcDcS7wSgzVi96GQqufR6tr2vPxOUv7H+8Z5q+MvWHTDTvw2b6rZfxrP8vQyhiD4JSfqjIvcQ7tiozYvDKBN4Gxz5vPQfUrT9KPATzsdMJbIBTfKlGugMwN" } }, { "type": "step", "result": "=\\frac{\\frac{1}{x}x^{6}-6x^{5}\\ln\\left(x\\right)}{x^{12}}" }, { "type": "interim", "title": "$$\\frac{1}{x}x^{6}=x^{5}$$", "input": "\\frac{1}{x}x^{6}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x^{6}}{x}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{6}=x^{6}$$", "result": "=\\frac{x^{6}}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=x^{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WVLhIKJxAHj/60V2C9gza9tK4+BSu6JNcWLSxs5NHVqjkVi15I8rBefLi4Iyt2wr+iK/i4AcYi8oiRmy8FYojmyKJewY067QC1vSYz7l2iF2VRbxoxPqXyVNbo8z3h7D" } }, { "type": "step", "result": "=\\frac{x^{5}-6x^{5}\\ln\\left(x\\right)}{x^{12}}" }, { "type": "step", "primary": "Factor out common term $$x^{5}$$", "result": "=\\frac{x^{5}\\left(1-6\\ln\\left(x\\right)\\right)}{x^{12}}", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Cancel $$\\frac{x^{5}\\left(1-6\\ln\\left(x\\right)\\right)}{x^{12}}:{\\quad}\\frac{1-6\\ln\\left(x\\right)}{x^{7}}$$", "input": "\\frac{x^{5}\\left(1-6\\ln\\left(x\\right)\\right)}{x^{12}}", "result": "=\\frac{1-6\\ln\\left(x\\right)}{x^{7}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{x^{5}}{x^{12}}=\\frac{1}{x^{12-5}}$$" ], "result": "=\\frac{1-6\\ln\\left(x\\right)}{x^{12-5}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$12-5=7$$", "result": "=\\frac{1-6\\ln\\left(x\\right)}{x^{7}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhQEtG+MAhknOlrmwiR27YhEnVxT22oCHnhtOAPD4DO63XeO2tIUPH5Q2xrCOU6NXcm10aTrmWMQZMPh736MBvkIJscNWPPx20gL5AX76ChPZEt3ZXAiqUE0HIXrrrezJEAWxP1tUPfnVZd59cq9pwAWhzJJwZUsfqAIiDNIXqkYhVS8+k7aJ6aCTXQq7wI964mpXFf3SOUx+H18qfp3MLg=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajaFatL1TR41CBlJ47CXHAp0cIhyz0QfD5ddZT+dhrxsPBt4LZAJ0vEoQQJEFkVkxRKuO77lnnV79llIPeBEfCMH5BSrUX+SGd9oFsj3gRn5hrpcRxMsRBQU8Dv0VSsGIZe9sGZu5A1MXROmEpnxG69ryk4whNiKT03wC0hzj1bDVaZR3lieU9xYVvi5Bke9mTHmMZgLxrnRkHNQMg9J5hH4RNcRA6FLeRMladeoJ8OE+aDkLa+SNXTXb5AQTFxkRZg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Solve $$\\frac{1-6\\ln\\left(x\\right)}{x^{7}}=0:{\\quad}x=\\sqrt[6]{e}$$", "input": "\\frac{1-6\\ln\\left(x\\right)}{x^{7}}=0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$x^{7}$$", "input": "\\frac{1-6\\ln\\left(x\\right)}{x^{7}}=0", "result": "1-6\\ln\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$x^{7}$$", "result": "\\frac{1-6\\ln\\left(x\\right)}{x^{7}}x^{7}=0\\cdot\\:x^{7}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1-6\\ln\\left(x\\right)}{x^{7}}x^{7}=0\\cdot\\:x^{7}", "result": "1-6\\ln\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{1-6\\ln\\left(x\\right)}{x^{7}}x^{7}:{\\quad}1-6\\ln\\left(x\\right)$$", "input": "\\frac{1-6\\ln\\left(x\\right)}{x^{7}}x^{7}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(1-6\\ln\\left(x\\right)\\right)x^{7}}{x^{7}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{7}$$", "result": "=1-6\\ln\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JFg64ksoQwIdHIf4YfSbCBdkkZUUOvJAWU8C+HK+Z/wAlilG71elit3w1IBbYN0P63OutcpoF1aeI/FWvgAk0WxKQKTSa9U4dlsoPwvj+RVN5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+YrCH7+gjQM+C3r/2uQ0KQ/sfKDjb9qmt0p77+Xy5lli" } }, { "type": "interim", "title": "Simplify $$0\\cdot\\:x^{7}:{\\quad}0$$", "input": "0\\cdot\\:x^{7}", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nFyi8TWqOpgY62prE5+GmS061ljBSPJeENOw2efoSWtRZPRrfkNDmi+szkABFipURSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rlQ4B6YwE+QTdHx0GZZ7zM" } }, { "type": "step", "result": "1-6\\ln\\left(x\\right)=0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1-6\\ln\\left(x\\right)=0", "result": "-6\\ln\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1-6\\ln\\left(x\\right)-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "-6\\ln\\left(x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-6$$", "input": "-6\\ln\\left(x\\right)=-1", "result": "\\ln\\left(x\\right)=\\frac{1}{6}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-6$$", "result": "\\frac{-6\\ln\\left(x\\right)}{-6}=\\frac{-1}{-6}" }, { "type": "step", "primary": "Simplify", "result": "\\ln\\left(x\\right)=\\frac{1}{6}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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$$b=a^c$$", "secondary": [ "$$\\ln\\left(x\\right)=\\frac{1}{6}\\quad\\:\\Rightarrow\\:\\quad\\:x=\\sqrt[6]{e}$$" ], "result": "x=\\sqrt[6]{e}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72g5TZdkJwY/HFcjVCNUrnz7tROVu4YtbQ32NyYqHihD1fWutlCU4zUzgkFchUVlK6qf7CVwD9kCy/9B7C1zLMM+aTjZPlHqPOg6v03oGo1MKCVVnZT+1Rrg2lstkW8Rv72wZm7kDUxdE6YSmfEbr2kK5TZnq5qTZnLZeS5lEw2KNmfj9J2TEf9Pt3Uz6EPr7" } }, { "type": "interim", "title": "Verify Solutions:$${\\quad}x=\\sqrt[6]{e}\\:$$True", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\frac{1-6\\ln\\left(x\\right)}{x^{7}}=0$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug in $$x=\\sqrt[6]{e}:{\\quad}$$True", "input": "\\frac{1-6\\ln\\left(\\sqrt[6]{e}\\right)}{\\left(\\sqrt[6]{e}\\right)^{7}}=0", "steps": [ { "type": "interim", "title": "$$\\frac{1-6\\ln\\left(\\sqrt[6]{e}\\right)}{\\left(\\sqrt[6]{e}\\right)^{7}}=0$$", "input": "\\frac{1-6\\ln\\left(\\sqrt[6]{e}\\right)}{\\left(\\sqrt[6]{e}\\right)^{7}}", "steps": [ { "type": "interim", "title": "$$\\left(\\sqrt[6]{e}\\right)^{7}=e^{\\frac{7}{6}}$$", "input": "\\left(\\sqrt[6]{e}\\right)^{7}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "result": "=\\left(e^{\\frac{1}{6}}\\right)^{7}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=e^{\\frac{1}{6}\\cdot\\:7}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{6}\\cdot\\:7=\\frac{7}{6}$$", "input": "\\frac{1}{6}\\cdot\\:7", "result": "=e^{\\frac{7}{6}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:7}{6}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:7=7$$", "result": "=\\frac{7}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZTPdCvP35NKbVf5YXx+lq6Uyhx2U/i+m6hs8FhqUqK+rju+5Z51e/ZZSD3gRHwjB3AqyH74BItUpRh7dqnvo1GRLd2VwIqlBNByF6663syRSCxTHoSWtnL6KZrH+w6IIcjNi7lbSR/+Acnt3fRtPxrCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wkvZH1wvncDcJlhZnNH1Slw6FQTjqOFY2dpMCUby1qnMwViaLUXkeD+JukROhWdjyijMHwavy9DgSTjaYfyN7P8//6/nV5O4fb8Xgwi7maroRLWThySq50+7tDvNiJNMqTctG2+xFiYYMAgFyWQTIFuJRCjLSfBWc/+7NRu8cQ0=" } }, { "type": "step", "result": "=\\frac{1-6\\ln\\left(\\sqrt[6]{e}\\right)}{e^{\\frac{7}{6}}}" }, { "type": "interim", "title": "$$6\\ln\\left(\\sqrt[6]{e}\\right)=1$$", "input": "6\\ln\\left(\\sqrt[6]{e}\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(\\sqrt[6]{e}\\right):{\\quad}\\frac{1}{6}$$", "input": "\\ln\\left(\\sqrt[6]{e}\\right)", "result": "=6\\cdot\\:\\frac{1}{6}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=\\ln\\left(e^{\\frac{1}{6}}\\right)" }, { "type": "step", "primary": "Apply log rule $$\\log_{a}\\left(x^b\\right)=b\\cdot\\log_{a}\\left(x\\right),\\:\\quad$$ assuming $$x\\:\\geq\\:0$$", "result": "=\\frac{1}{6}\\ln\\left(e\\right)" }, { "type": "step", "primary": "Apply log rule: $$\\log_a\\left(a\\right)=1$$", "result": "=\\frac{1}{6}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:6}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$6$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7etJBfUGnnojN3p6Ur4AOLyuTZ7J6vh3+nLtlhtm6cDtwkKGJWEPFPk38sdJMsyPIeqXfySbC6vm4UawE43QWXbZ+V4IzZEEbaTtpN5sPgsxFuyFcX5GALo7VXRtjO0PT" } }, { "type": "step", "result": "=\\frac{1-1}{e^{\\frac{7}{6}}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1-1}{e^{\\frac{7}{6}}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$1-1=0$$", "result": "=\\frac{0}{e^{\\frac{7}{6}}}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver" } }, { "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": "x=\\sqrt[6]{e}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Find undefined (singularity) points of $$f^{\\prime}\\left(x\\right):{\\quad}x=0$$", "steps": [ { "type": "interim", "title": "Undefined points for $$\\frac{1}{f\\left(x\\right)}:{\\quad}x=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{1-6\\ln\\left(x\\right)}{x^{7}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{7}=0:{\\quad}x=0$$", "input": "x^{7}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points Div0 0Eq" } }, { "type": "interim", "title": "Undefined points for $$\\log\\left(f\\left(x\\right)\\right):{\\quad}x=0$$", "steps": [ { "type": "step", "primary": "Take the content of the log expression(s) of $$\\frac{1-6\\ln\\left(x\\right)}{x^{7}}$$ and compare to zero" }, { "type": "step", "result": "x=0" }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points Log0 0Eq" } }, { "type": "step", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "result": "x=\\sqrt[6]{e},\\:x=0" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/x1nUSsx/WhOV8CTUVRVLWnbaogaJwXE7SiNkAcy1aVy/HI5S0StY1FdtOqqOPr0TeyvDofYUUcGQ3+3ntDzllMGB5gMbVaYSGu0dQEN45QHlkQhNa5OD0svv0fHOCAFzxj/L0MoYg+CUn6oyL3EO7YpU5sZoQmlpjZz67pEMqqgaBDX9aXlgA5Wc5b7KLVldnTFK52Ct5rw3AqKJ1lrCqIekOAO0oWIUqTczTA8wjRnTpGxMf4x+E8pX+LLr4nZsBg==" } }, { "type": "step", "primary": "Identify critical points not in $$f\\left(x\\right)$$ domain" }, { "type": "interim", "title": "Domain of $$\\frac{\\ln\\left(x\\right)}{x^{6}}\\::{\\quad}x>0$$", "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": "interim", "title": "Find positive values for logs:$${\\quad}x>0$$", "input": "\\frac{\\ln\\left(x\\right)}{x^{6}}", "steps": [ { "type": "step", "primary": "$$\\log_a{f\\left(x\\right)}\\quad\\Rightarrow\\quad\\:f\\left(x\\right)>0$$", "meta": { "general_rule": { "extension": "$$\\log_a{f\\left(x\\right)}$$ has real values only when $$f\\left(x\\right)>0$$" } } }, { "type": "step", "result": "x>0" } ], "meta": { "interimType": "Positive Logs 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6nqFF93wFkmFLIZ1zIJ4nmNRd3zuucjSjqZ/0vlC0vJap+eMFXj7VmBe5iHljtk6ggrQRYup/iWv53iA1tMV70Cjeh7+jKEzLb7VNCEMF3Z/Q+lnlzyyO0IqCE9Su0AXgqu3Iduefr1w2DEFJVRhcrOfs+V6JshXkV/96oCfzSN+" } }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=0$$", "input": "\\frac{\\ln\\left(x\\right)}{x^{6}}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{\\ln\\left(x\\right)}{x^{6}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x^{6}=0:{\\quad}x=0$$", "input": "x^{6}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=0" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6nqFF93wFkmFLIZ1zIJ4nmPyxWRGnnGINMUXH6CHfF7/M6/W7rkloBMm6OqTQ9UW9CxOUtrYRq4BvNNr2XUu6o5NuEM6hxBraOt7lssmGP08RSpN33oxZMojoqvYhvSJAM7smHY5JYb8vLrutydV3LMVJfwlOggS7xEa0KlmyHCxxq714IG8qF7AXh5s7M1Yyg==" } }, { "type": "step", "primary": "Combine real regions and undefined points for final function domain", "result": "x>0" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "step", "primary": "$$\\frac{\\ln\\left(x\\right)}{x^{6}}$$ is not defined at $$x=0$$, therefore:", "result": "x=\\sqrt[6]{e}" } ], "meta": { "solvingClass": "Function Critical" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{\\ln(x)}{x^{6}}" }, "showViewLarger": true } } }