monotone x^3-x^2+2x-1

{ "query": { "display": "monotone intervals $$x^{3}-x^{2}+2x-1$$", "symbolab_question": "FUNCTION#monotone x^{3}-x^{2}+2x-1" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "monotone", "default": "\\mathrm{Increasing}:-\\infty <x<\\infty ", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Monotone Intervals of $$x^{3}-x^{2}+2x-1:{\\quad}$$Increasing$$:-\\infty\\:<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Monotone intervals definition", "text": "If $$f'\\left(x\\right)>0\\:$$then $$f\\left(x\\right)\\:$$is increasing.<br/>If $$f'\\left(x\\right)<0\\:$$then $$f\\left(x\\right)\\:$$is decreasing." }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=3x^{2}-2x+2$$", "input": "\\frac{d}{dx}\\left(x^{3}-x^{2}+2x-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{3}\\right)-\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(2x\\right)-\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{3}\\right)=3x^{2}$$", "input": "\\frac{d}{dx}\\left(x^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtb6j95rHG7YtZ73Xx2qCjqk3hxk9aCfAWodBRxXgUexf7nh0v5ML3fMP9GgRVbRX/8//6/nV5O4fb8Xgwi7maqO95Is7YBdcRqXmTH8euc2MRY7LLv3QukQErzdJ9wdtQ==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=3x^{2}-2x+2-0" }, { "type": "step", "primary": "Simplify", "result": "=3x^{2}-2x+2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Increasing$$:-\\infty\\:<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=3x^{2}-2x+2", "steps": [ { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}$$True for all $$x\\in\\mathbb{R}$$", "input": "3x^{2}-2x+2>0", "steps": [ { "type": "interim", "title": "Complete the square $$3x^{2}-2x+2:{\\quad}3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}$$", "input": "3x^{2}-2x+2", "steps": [ { "type": "step", "primary": "Write $$3x^{2}-2x+2\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$3$$" ], "result": "3\\left(x^{2}-\\frac{2x}{3}+\\frac{2}{3}\\right)" }, { "type": "interim", "title": "$$2a=-\\frac{2}{3}{\\quad:\\quad}a=-\\frac{1}{3}$$", "input": "2a=-\\frac{2}{3}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-\\frac{2}{3}", "result": "a=-\\frac{1}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-\\frac{2}{3}}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2a}{2}=\\frac{-\\frac{2}{3}}{2}", "result": "a=-\\frac{1}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2a}{2}:{\\quad}a$$", "input": "\\frac{2a}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=a" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n/9TIosDfOyZXH4usT9jsy061ljBSPJeENOw2efoSWuFFDuzzgIzA1F36c+SAFGgRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pQyGeXuoJHAaVrX92ZjOTT" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\frac{2}{3}}{2}:{\\quad}-\\frac{1}{3}$$", "input": "\\frac{-\\frac{2}{3}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{2}{3}}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{2}{3}}{2}=\\frac{2}{3\\cdot\\:2}$$" ], "result": "=-\\frac{2}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=-\\frac{2}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVTiHmrOIjI+3CDvd17UTeibNGoPE9TME3q+OPmgkv2RQrRKMQ0pA8A67jlmbMMkBF/cj3oMDrtAdIlhRv6tbzHyY3ASC+aZqPN1DBWUUsybFA5wCrLmSQJycqzxqW5H7bsuMBQWmWJ6ECMfmHytwiOy/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "a=-\\frac{1}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-\\frac{1}{3}\\right)^{2}\\:$$", "result": "3\\left(x^{2}-\\frac{2x}{3}+\\frac{2}{3}+\\left(-\\frac{1}{3}\\right)^{2}-\\left(-\\frac{1}{3}\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-\\frac{2}{3}x+\\left(-\\frac{1}{3}\\right)^{2}=\\left(x-\\frac{1}{3}\\right)^{2}$$", "Complete the square" ], "result": "3\\left(\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{2}{3}-\\left(-\\frac{1}{3}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}>0" }, { "type": "interim", "title": "Move $$\\frac{5}{3}\\:$$to the right side", "input": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}>0", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}>-\\frac{5}{3}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{5}{3}$$ from both sides", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}-\\frac{5}{3}>0-\\frac{5}{3}" }, { "type": "step", "primary": "Simplify", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}>-\\frac{5}{3}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3\\left(x-\\frac{1}{3}\\right)^{2}>-\\frac{5}{3}", "result": "\\left(x-\\frac{1}{3}\\right)^{2}>-\\frac{5}{9}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}>\\frac{-\\frac{5}{3}}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}>\\frac{-\\frac{5}{3}}{3}", "result": "\\left(x-\\frac{1}{3}\\right)^{2}>-\\frac{5}{9}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}:{\\quad}\\left(x-\\frac{1}{3}\\right)^{2}$$", "input": "\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=\\left(x-\\frac{1}{3}\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73bSiKscGPD/sQfmK5EacDsJXLlvGDZ2xxeBeV16cVCXk9CU8MCWpvAxSxvbI1SlDAGrABULuPgjgJWRoN7CtxB4806N8LVUIZNOkrWClWdJeTZMykILbMC5S4vTIC/oK7kAjP76qW66lOUsURwT0ndafV1nrpj6r2YbQTKXwLq71rbTYQy6tN5N5Y1iznpKn7DCYUL/JLb2U/ppdPh1KXg==" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\frac{5}{3}}{3}:{\\quad}-\\frac{5}{9}$$", "input": "\\frac{-\\frac{5}{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{5}{3}}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{5}{3}}{3}=\\frac{5}{3\\cdot\\:3}$$" ], "result": "=-\\frac{5}{3\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=-\\frac{5}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVbk3Qyfz8lvKh7IY3nP5Pc/NGoPE9TME3q+OPmgkv2RQx9bx+VaFnjuxyD9q5O2WEajfdIOMuQQNq+PUtNxnqhuY3ASC+aZqPN1DBWUUsybFA5wCrLmSQJycqzxqW5H7bjiFLuXK6brNgBeXbCmgFH2/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "\\left(x-\\frac{1}{3}\\right)^{2}>-\\frac{5}{9}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "If n is even, $$u^n\\:\\geq\\:0$$ for all $$u$$", "result": "\\mathrm{True\\:for\\:all}\\:x" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}$$False for all $$x\\in\\mathbb{R}$$", "input": "3x^{2}-2x+2<0", "steps": [ { "type": "interim", "title": "Complete the square $$3x^{2}-2x+2:{\\quad}3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}$$", "input": "3x^{2}-2x+2", "steps": [ { "type": "step", "primary": "Write $$3x^{2}-2x+2\\:$$in the form: $$x^2+2bx+b^2$$", "secondary": [ "Factor out $$3$$" ], "result": "3\\left(x^{2}-\\frac{2x}{3}+\\frac{2}{3}\\right)" }, { "type": "interim", "title": "$$2b=-\\frac{2}{3}{\\quad:\\quad}b=-\\frac{1}{3}$$", "input": "2b=-\\frac{2}{3}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2b=-\\frac{2}{3}", "result": "b=-\\frac{1}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2b}{2}=\\frac{-\\frac{2}{3}}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2b}{2}=\\frac{-\\frac{2}{3}}{2}", "result": "b=-\\frac{1}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2b}{2}:{\\quad}b$$", "input": "\\frac{2b}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=b" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wbPe5/kMaTHYl8T/JSLt7C061ljBSPJeENOw2efoSWunm6UsOWrJA3xF0Dzdl6zQRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pXKKmv5DfDZus0moEuQZbo" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\frac{2}{3}}{2}:{\\quad}-\\frac{1}{3}$$", "input": "\\frac{-\\frac{2}{3}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{2}{3}}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{2}{3}}{2}=\\frac{2}{3\\cdot\\:2}$$" ], "result": "=-\\frac{2}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=-\\frac{2}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVTiHmrOIjI+3CDvd17UTeibNGoPE9TME3q+OPmgkv2RQrRKMQ0pA8A67jlmbMMkBF/cj3oMDrtAdIlhRv6tbzHyY3ASC+aZqPN1DBWUUsybFA5wCrLmSQJycqzxqW5H7bsuMBQWmWJ6ECMfmHytwiOy/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "b=-\\frac{1}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-\\frac{1}{3}\\right)^{2}\\:$$", "result": "3\\left(x^{2}-\\frac{2x}{3}+\\frac{2}{3}+\\left(-\\frac{1}{3}\\right)^{2}-\\left(-\\frac{1}{3}\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2bx+b^2=\\left(x+b\\right)^2$$", "secondary": [ "$$x^{2}-\\frac{2}{3}x+\\left(-\\frac{1}{3}\\right)^{2}=\\left(x-\\frac{1}{3}\\right)^{2}$$", "Complete the square" ], "result": "3\\left(\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{2}{3}-\\left(-\\frac{1}{3}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}<0" }, { "type": "interim", "title": "Move $$\\frac{5}{3}\\:$$to the right side", "input": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}<0", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}<-\\frac{5}{3}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{5}{3}$$ from both sides", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}+\\frac{5}{3}-\\frac{5}{3}<0-\\frac{5}{3}" }, { "type": "step", "primary": "Simplify", "result": "3\\left(x-\\frac{1}{3}\\right)^{2}<-\\frac{5}{3}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3\\left(x-\\frac{1}{3}\\right)^{2}<-\\frac{5}{3}", "result": "\\left(x-\\frac{1}{3}\\right)^{2}<-\\frac{5}{9}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}<\\frac{-\\frac{5}{3}}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}<\\frac{-\\frac{5}{3}}{3}", "result": "\\left(x-\\frac{1}{3}\\right)^{2}<-\\frac{5}{9}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}:{\\quad}\\left(x-\\frac{1}{3}\\right)^{2}$$", "input": "\\frac{3\\left(x-\\frac{1}{3}\\right)^{2}}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=\\left(x-\\frac{1}{3}\\right)^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73bSiKscGPD/sQfmK5EacDsJXLlvGDZ2xxeBeV16cVCXk9CU8MCWpvAxSxvbI1SlDAGrABULuPgjgJWRoN7CtxB4806N8LVUIZNOkrWClWdJeTZMykILbMC5S4vTIC/oK7kAjP76qW66lOUsURwT0ndafV1nrpj6r2YbQTKXwLq71rbTYQy6tN5N5Y1iznpKn7DCYUL/JLb2U/ppdPh1KXg==" } }, { "type": "interim", "title": "Simplify $$\\frac{-\\frac{5}{3}}{3}:{\\quad}-\\frac{5}{9}$$", "input": "\\frac{-\\frac{5}{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{5}{3}}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{5}{3}}{3}=\\frac{5}{3\\cdot\\:3}$$" ], "result": "=-\\frac{5}{3\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=-\\frac{5}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78faTMm53GPovDqGWXYdIVbk3Qyfz8lvKh7IY3nP5Pc/NGoPE9TME3q+OPmgkv2RQx9bx+VaFnjuxyD9q5O2WEajfdIOMuQQNq+PUtNxnqhuY3ASC+aZqPN1DBWUUsybFA5wCrLmSQJycqzxqW5H7bjiFLuXK6brNgBeXbCmgFH2/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "\\left(x-\\frac{1}{3}\\right)^{2}<-\\frac{5}{9}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TyACwKQn6tnWli5mA8M13IIFmpNCiY40/iVRZ/JAmsztAzGeWeWnKmgbIGTT1eJMdCBWanF6W9xxkKeItJVTcAhTQ7NPxL7UwD055yztwZYUbV6iKojeivOTyyef34syJPmI6/q9/PFesHOJhM+nIj+KGm35uF0U7WjdCvywXwrI1elmnb8ymdcGud2ePUarU9ik5WINGrsWotLzS7/BjMilG2DUqMrP9Xo55x7KtrDuMk2xuPJ9Xrfj77W2cr6YQCdy2hEwcw+SiSeOW6aUMDGZPKNiTwhEcLC3SmJs5DIW3/5twYA89w9TwRhdo2lZS3D4Die0wB8SlCsv2r53JRWTy3ZPCOH7OOn12VhNyj8aDBJshx3kmM/93Uq4Jtb3jMNJ/a+H+L7ezhw7+grBnLWsrD7vZXjUuDF4I55wpc58i+1EUi0JR87vYNuUkO+vpqDfn2ckgJI3FHWx2SIlgPC30sSftAIFS6Qkpy19IkrRZxLY5D4o5yBtIEUXy0AtYS99FT8QV1ZcS9eXGZoiAA==" } }, { "type": "step", "primary": "If n is even, $$u^n\\:\\ge\\:0\\:$$ for all $$u$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "-\\infty\\:<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$x^{3}-x^{2}+2x-1\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Combine $$-\\infty\\:<x<\\infty\\:\\:$$ with domain:$${\\quad}-\\infty\\:<x<\\infty\\:$$", "input": "-\\infty\\:<x<\\infty\\:\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "step", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "interimType": "Combine Intervals With Domain 0Eq" } }, { "type": "step", "primary": "Summary of the monotone intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|}\\hline &-\\infty <x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime}}(x)>0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Increasing}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Increasing}:-\\infty\\:<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQnSGLMy6w719XZab5nwoBBKSZnesI6ClS1+W9Js2+Na8dMS4tdBTb6vw56PKDresHfTCz0/OgpozJTcCVZ2t6Qf/C6bOUY9eLMGqMHppqcIw=" } }, { "type": "step", "result": "\\mathrm{Increasing}:-\\infty\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Monotone" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{3}-x^{2}+2x-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }