{
"query": {
"display": "monotone intervals $$f\\left(x\\right)=x-3\\sqrt[3]{x}$$",
"symbolab_question": "FUNCTION#monotone f(x)=x-3\\sqrt[3]{x}"
},
"solution": {
"level": "PERFORMED",
"subject": "Functions & Graphing",
"topic": "Functions",
"subTopic": "monotone",
"default": "\\mathrm{Decreasing}:-\\infty <x<-1,\\mathrm{Increasing}:-1<x<1,\\mathrm{Decreasing}:1<x<\\infty ",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "Monotone Intervals of $$x-3\\sqrt[3]{x}:{\\quad}$$Decreasing$$:-\\infty\\:<x<-1,\\:$$Increasing$$:-1<x<1,\\:$$Decreasing$$:1<x<\\infty\\:$$",
"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": "$$f\\:{^{\\prime}}\\left(x\\right)=1-\\frac{1}{x^{\\frac{2}{3}}}$$",
"input": "\\frac{d}{dx}\\left(x-3\\sqrt[3]{x}\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(3\\sqrt[3]{x}\\right)"
},
{
"type": "interim",
"title": "$$\\frac{dx}{dx}=1$$",
"input": "\\frac{dx}{dx}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(3\\sqrt[3]{x}\\right)=\\frac{1}{x^{\\frac{2}{3}}}$$",
"input": "\\frac{d}{dx}\\left(3\\sqrt[3]{x}\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=3\\frac{d}{dx}\\left(\\sqrt[3]{x}\\right)"
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$",
"result": "=3\\frac{d}{dx}\\left(x^{\\frac{1}{3}}\\right)",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$",
"result": "=3\\cdot\\:\\frac{1}{3}x^{\\frac{1}{3}-1}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule",
"practiceTopic": "Power Rule"
}
},
{
"type": "interim",
"title": "Simplify $$3\\cdot\\:\\frac{1}{3}x^{\\frac{1}{3}-1}:{\\quad}\\frac{1}{x^{\\frac{2}{3}}}$$",
"input": "3\\cdot\\:\\frac{1}{3}x^{\\frac{1}{3}-1}",
"result": "=\\frac{1}{x^{\\frac{2}{3}}}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:3x^{\\frac{1}{3}-1}}{3}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$3$$",
"result": "=1\\cdot\\:x^{\\frac{1}{3}-1}"
},
{
"type": "interim",
"title": "$$x^{\\frac{1}{3}-1}=x^{-\\frac{2}{3}}$$",
"input": "x^{\\frac{1}{3}-1}",
"steps": [
{
"type": "interim",
"title": "Join $$\\frac{1}{3}-1:{\\quad}-\\frac{2}{3}$$",
"input": "\\frac{1}{3}-1",
"result": "=x^{-\\frac{2}{3}}",
"steps": [
{
"type": "step",
"primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$",
"result": "=-\\frac{1\\cdot\\:3}{3}+\\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{-1\\cdot\\:3+1}{3}"
},
{
"type": "interim",
"title": "$$-1\\cdot\\:3+1=-2$$",
"input": "-1\\cdot\\:3+1",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$1\\cdot\\:3=3$$",
"result": "=-3+1"
},
{
"type": "step",
"primary": "Add/Subtract the numbers: $$-3+1=-2$$",
"result": "=-2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iSe/oPZ15XQklhwmxDEIoVXTSum/z5kLpMzXS1UJIexttTTHAByi5+zcJ9JCN6ZAyCE30rzMlUAIVDyhseMBrufcnznbZfVbgKLGaldkjXU="
}
},
{
"type": "step",
"result": "=\\frac{-2}{3}"
},
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$",
"result": "=-\\frac{2}{3}"
}
],
"meta": {
"interimType": "Algebraic Manipulation Join Concise Title 1Eq"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DborvwU589/20W5rwUAp2+0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjeVW9/X7jmVoO3IMaXztB2wH2kDe5DGYTz3TrPquGdIhR/IWXCqoVxn0KSq5yq0Z+6M8osviUPEkWv33aMbZrSIBcZlU2JZxWsEokVU5V0NU="
}
},
{
"type": "step",
"result": "=1\\cdot\\:x^{-\\frac{2}{3}}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$",
"secondary": [
"$$x^{-\\frac{2}{3}}=\\frac{1}{x^{\\frac{2}{3}}}$$"
],
"result": "=1\\cdot\\:\\frac{1}{x^{\\frac{2}{3}}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{\\frac{2}{3}}}=\\frac{1}{x^{\\frac{2}{3}}}$$",
"result": "=\\frac{1}{x^{\\frac{2}{3}}}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFAxuAhzfH7ZbhM8JXdppsXF3jt0YWJlZUdkIYOP42z9IldYPfXQvX4/bINBB8wSEQ0U1Y7SwT225b4rPxVIUQTzgnA6BN4Zi/43DUpsJDK6CbP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXRazroBYpdJ8ysZOU8vZ64sQPzAZEvjTNZd6Dsb74kOlDGnaEkdXzmzeDi9NT6Lpkg="
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=1-\\frac{1}{x^{\\frac{2}{3}}}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "Find intervals:$${\\quad}$$Decreasing$$:-\\infty\\:<x<-1,\\:$$Increasing$$:-1<x<1,\\:$$Decreasing$$:1<x<\\infty\\:$$",
"input": "f\\:{^{\\prime}}\\left(x\\right)=1-\\frac{1}{x^{\\frac{2}{3}}}",
"steps": [
{
"type": "interim",
"title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}-1<x<1$$",
"input": "1-\\frac{1}{x^{\\frac{2}{3}}}>0",
"steps": [
{
"type": "interim",
"title": "Move $$1\\:$$to the right side",
"input": "1-\\left(\\frac{1}{x^{\\frac{2}{3}}}\\right)>0",
"result": "-\\frac{1}{x^{\\frac{2}{3}}}>-1",
"steps": [
{
"type": "step",
"primary": "Subtract $$1$$ from both sides",
"result": "1-\\frac{1}{x^{\\frac{2}{3}}}-1>0-1"
},
{
"type": "step",
"primary": "Simplify",
"result": "-\\frac{1}{x^{\\frac{2}{3}}}>-1"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
},
{
"type": "interim",
"title": "Multiply both sides by $$-1$$",
"input": "-\\left(\\frac{1}{x^{\\frac{2}{3}}}\\right)>-1",
"result": "\\frac{1}{x^{\\frac{2}{3}}}<1",
"steps": [
{
"type": "step",
"primary": "Multiply both sides by -1 (reverse the inequality)",
"result": "\\left(-\\frac{1}{x^{\\frac{2}{3}}}\\right)\\left(-1\\right)<\\left(-1\\right)\\left(-1\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "\\frac{1}{x^{\\frac{2}{3}}}<1"
}
],
"meta": {
"interimType": "Multiply Both Sides Specific 1Eq",
"gptData": "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"
}
},
{
"type": "step",
"primary": "If$$\\quad\\:\\frac{1}{a}<0\\:\\quad$$then$$\\quad\\:a<0$$",
"result": "x^{\\frac{2}{3}}<1"
},
{
"type": "step",
"primary": "Take both sides of the equation to the power of $$3$$",
"result": "\\left(x^{\\frac{2}{3}}\\right)^{3}<1^{3}"
},
{
"type": "step",
"primary": "Simplify",
"result": "x^{2}<1"
},
{
"type": "interim",
"title": "$$x^{2}<1{\\quad:\\quad}-1<x<1$$",
"input": "x^{2}<1",
"steps": [
{
"type": "step",
"primary": "For $$u^n\\:<\\:a$$, if $$n\\:$$is even then $$-\\sqrt[n]{a}\\:<\\:u\\:<\\:\\sqrt[n]{a}$$"
},
{
"type": "step",
"result": "-1<x<1"
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
},
{
"type": "step",
"result": "-1<x<1"
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
},
{
"type": "interim",
"title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}x<-1\\lor\\:x>1$$",
"input": "1-\\frac{1}{x^{\\frac{2}{3}}}<0",
"steps": [
{
"type": "interim",
"title": "Move $$1\\:$$to the right side",
"input": "1-\\left(\\frac{1}{x^{\\frac{2}{3}}}\\right)<0",
"result": "-\\frac{1}{x^{\\frac{2}{3}}}<-1",
"steps": [
{
"type": "step",
"primary": "Subtract $$1$$ from both sides",
"result": "1-\\frac{1}{x^{\\frac{2}{3}}}-1<0-1"
},
{
"type": "step",
"primary": "Simplify",
"result": "-\\frac{1}{x^{\\frac{2}{3}}}<-1"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
},
{
"type": "interim",
"title": "Multiply both sides by $$-1$$",
"input": "-\\left(\\frac{1}{x^{\\frac{2}{3}}}\\right)<-1",
"result": "\\frac{1}{x^{\\frac{2}{3}}}>1",
"steps": [
{
"type": "step",
"primary": "Multiply both sides by -1 (reverse the inequality)",
"result": "\\left(-\\frac{1}{x^{\\frac{2}{3}}}\\right)\\left(-1\\right)>\\left(-1\\right)\\left(-1\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "\\frac{1}{x^{\\frac{2}{3}}}>1"
}
],
"meta": {
"interimType": "Multiply Both Sides Specific 1Eq",
"gptData": "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"
}
},
{
"type": "step",
"primary": "If$$\\quad\\:\\frac{1}{a}>0\\:\\quad$$then$$\\quad\\:a>0$$",
"result": "x^{\\frac{2}{3}}>1"
},
{
"type": "step",
"primary": "Take both sides of the equation to the power of $$3$$",
"result": "\\left(x^{\\frac{2}{3}}\\right)^{3}>1^{3}"
},
{
"type": "step",
"primary": "Simplify",
"result": "x^{2}>1"
},
{
"type": "interim",
"title": "$$x^{2}>1{\\quad:\\quad}x<-1\\lor\\:x>1$$",
"input": "x^{2}>1",
"steps": [
{
"type": "step",
"primary": "For $$u^n\\:>\\:a$$, if $$n\\:$$is even then $$u\\:<\\:-\\sqrt[n]{a}\\:or\\:u\\:>\\:\\sqrt[n]{a}$$"
},
{
"type": "step",
"result": "x<-1\\lor\\:x>1"
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
},
{
"type": "step",
"result": "x<-1\\lor\\:x>1"
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
},
{
"type": "interim",
"title": "Combine intervals with domain",
"result": "-\\infty\\:<x<-1,\\:-1<x<1,\\:1<x<\\infty\\:",
"steps": [
{
"type": "interim",
"title": "Domain of $$x-3\\sqrt[3]{x}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$",
"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 $$-1<x<1\\:$$ with domain:$${\\quad}-1<x<1$$",
"input": "-1<x<1\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}",
"steps": [
{
"type": "step",
"primary": "Simplify",
"result": "-1<x<1"
}
],
"meta": {
"interimType": "Combine With Domain 1Eq"
}
},
{
"type": "interim",
"title": "Combine $$-\\infty\\:<x<-1\\:$$ with domain:$${\\quad}-\\infty\\:<x<-1$$",
"input": "-\\infty\\:<x<-1\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}",
"steps": [
{
"type": "step",
"primary": "Simplify",
"result": "-\\infty\\:<x<-1"
}
],
"meta": {
"interimType": "Combine With Domain 1Eq"
}
},
{
"type": "interim",
"title": "Combine $$1<x<\\infty\\:\\:$$ with domain:$${\\quad}1<x<\\infty\\:$$",
"input": "1<x<\\infty\\:\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}",
"steps": [
{
"type": "step",
"primary": "Simplify",
"result": "1<x<\\infty\\:"
}
],
"meta": {
"interimType": "Combine With Domain 1Eq"
}
},
{
"type": "step",
"result": "-\\infty\\:<x<-1,\\:-1<x<1,\\:1<x<\\infty\\:"
}
],
"meta": {
"interimType": "Combine Intervals With Domain 0Eq"
}
},
{
"type": "step",
"primary": "Summary of the monotone intervals behavior",
"secondary": [
"$$\\begin{array}{|c|c|c|c|}\\hline &-\\infty <x<-1&-1<x<1&1<x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime}}(x)<0&f {^{\\prime}}(x)>0&f {^{\\prime}}(x)<0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Decreasing}&\\mathrm{Increasing}&\\mathrm{Decreasing}\\\\\\hline \\end{array}$$"
]
},
{
"type": "step",
"result": "\\mathrm{Decreasing}:-\\infty\\:<x<-1,\\:\\mathrm{Increasing}:-1<x<1,\\:\\mathrm{Decreasing}:1<x<\\infty\\:"
}
],
"meta": {
"interimType": "Function Find Intervals 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQnSGLMy6w719XZab5nwoBBEvCa/MJxTMLO4Mho2HIsusnA6BN4Zi/43DUpsJDK6Cb7FQ6cqE6LIN/CnlM5t3zzHimqO4VtL6M91XE4IPuJJuZHcHmragI9YnQ8UOMAymz"
}
},
{
"type": "step",
"result": "\\mathrm{Decreasing}:-\\infty\\:<x<-1,\\:\\mathrm{Increasing}:-1<x<1,\\:\\mathrm{Decreasing}:1<x<\\infty\\:"
}
],
"meta": {
"solvingClass": "Function Monotone"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "x-3\\sqrt[3]{x}"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solution
monotone intervals
Solution
Solution steps
Find intervals:DecreasingIncreasingDecreasing