extreme f(x)=x^4+2x^2

{ "query": { "display": "extreme points $$f\\left(x\\right)=x^{4}+2x^{2}$$", "symbolab_question": "FUNCTION#extreme f(x)=x^{4}+2x^{2}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "extreme", "default": "\\mathrm{Minimum}(0,0)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find using the First Derivative Test", "query": { "display": "first derivative test $$x^{4}+2x^{2}$$", "symbolab_question": "firstderivativetest x^{4}+2x^{2}" } }, { "method": "Find using the Second Derivative Test", "query": { "display": "second derivative test $$x^{4}+2x^{2}$$", "symbolab_question": "secondderivativetest x^{4}+2x^{2}" } } ], "steps": { "type": "interim", "title": "Extreme Points of $$x^{4}+2x^{2}:{\\quad}$$Minimum$$\\left(0,\\:0\\right)$$", "steps": [ { "type": "definition", "title": "First Derivative Test definition", "text": "Suppose that $$x=c$$ is a critical point of $$f\\left(x\\right)$$ then, <br/>If $$f\\:{^{\\prime}}\\left(x\\right)>0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the right of $$x=c$$ then $$x=c$$ is a local maximum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)>\\:0$$ to the right of $$x=c$$ then $$x=c$$ is a local minimum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)$$ is the same sign on both sides of $$x=c$$ then $$x=c$$ is neither a local maximum nor a local minimum." }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=4x^{3}+4x$$", "input": "\\frac{d}{dx}\\left(x^{4}+2x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{4}\\right)+\\frac{d}{dx}\\left(2x^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{4}\\right)=4x^{3}$$", "input": "\\frac{d}{dx}\\left(x^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4x^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4x^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYodvM0LC1fPom5KofzCj+6qk3hxk9aCfAWodBRxXgUexGgZz1CFzF7HTa4VF2uoRHv8//6/nV5O4fb8Xgwi7maqO95Is7YBdcRqXmTH8euc2JrcUvyUpj++aXrGYPlvDVw==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x^{2}\\right)=4x$$", "input": "\\frac{d}{dx}\\left(2x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgEShBvy1snibnAN3NvB1a2TdaV09PMxEKZ9FieghTFwHBO3D9VaGp1eOVvjTiCiEaN6Hv6MoTMtvtU0IQwXdn+XNwOQ43NHE8cpERrPgoqpfTZuddhTh3r/FmyVu4x1Bw==" } }, { "type": "step", "result": "=4x^{3}+4x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Decreasing$$:-\\infty\\:<x<0,\\:$$Increasing$$:0<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=4x^{3}+4x", "steps": [ { "type": "interim", "title": "Find the critical points:$${\\quad}x=0$$", "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": "$$f\\:{^{\\prime}}\\left(x\\right)=0:{\\quad}x=0$$", "input": "4x^{3}+4x=0", "steps": [ { "type": "interim", "title": "Factor $$4x^{3}+4x:{\\quad}4x\\left(x^{2}+1\\right)$$", "input": "4x^{3}+4x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{3}=x^{2}x$$" ], "result": "=4x^{2}x+4x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$4x$$", "result": "=4x\\left(x^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "4x\\left(x^{2}+1\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "x=0\\lor\\:x^{2}+1=0" }, { "type": "interim", "title": "Solve $$x^{2}+1=0:{\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "x^{2}+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x^{2}+1=0", "result": "x^{2}=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x^{2}+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=0" } ], "meta": { "interimType": "Explore Function Slope Zero Title 0Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}x>0$$", "input": "4x^{3}+4x>0", "steps": [ { "type": "interim", "title": "Rewrite in standard form", "input": "4x^{3}+4x>0", "result": "x^{3}+x>0", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4x^{3}}{4}+\\frac{4x}{4}>\\frac{0}{4}" }, { "type": "step", "primary": "Simplify", "result": "x^{3}+x>0", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$x^{3}+x:{\\quad}x\\left(x^{2}+1\\right)$$", "input": "x^{3}+x", "result": "x\\left(x^{2}+1\\right)>0", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{3}=x^{2}x$$" ], "result": "=x^{2}x+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Identify the intervals", "result": "x>0", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$x\\left(x^{2}+1\\right)$$" }, { "type": "interim", "title": "Find the signs of $$x$$", "steps": [ { "type": "step", "result": "x=0" }, { "type": "step", "result": "x<0" }, { "type": "step", "result": "x>0" } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x^{2}+1:{\\quad}$$Positive for all real values", "input": "x^{2}+1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x^{2}+1>0", "result": "x^{2}>-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x^{2}+1-1>0-1" }, { "type": "step", "primary": "Simplify", "result": "x^{2}>-1" } ], "meta": { "interimType": "Move to the Right Title 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": "Find Sign 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|}\\hline &x<0&x=0&x>0\\\\\\hline x&-&0&+\\\\\\hline x^{2}+1&+&+&+\\\\\\hline x(x^{2}+1)&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$>\\:0$$", "result": "x>0" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}x<0$$", "input": "4x^{3}+4x<0", "steps": [ { "type": "interim", "title": "Rewrite in standard form", "input": "4x^{3}+4x<0", "result": "x^{3}+x<0", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4x^{3}}{4}+\\frac{4x}{4}<\\frac{0}{4}" }, { "type": "step", "primary": "Simplify", "result": "x^{3}+x<0", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$x^{3}+x:{\\quad}x\\left(x^{2}+1\\right)$$", "input": "x^{3}+x", "result": "x\\left(x^{2}+1\\right)<0", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{3}=x^{2}x$$" ], "result": "=x^{2}x+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Identify the intervals", "result": "x<0", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$x\\left(x^{2}+1\\right)$$" }, { "type": "interim", "title": "Find the signs of $$x$$", "steps": [ { "type": "step", "result": "x=0" }, { "type": "step", "result": "x<0" }, { "type": "step", "result": "x>0" } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x^{2}+1:{\\quad}$$Positive for all real values", "input": "x^{2}+1>0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x^{2}+1>0", "result": "x^{2}>-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "x^{2}+1-1>0-1" }, { "type": "step", "primary": "Simplify", "result": "x^{2}>-1" } ], "meta": { "interimType": "Move to the Right Title 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": "Find Sign 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|}\\hline &x<0&x=0&x>0\\\\\\hline x&-&0&+\\\\\\hline x^{2}+1&+&+&+\\\\\\hline x(x^{2}+1)&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$<\\:0$$", "result": "x<0" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "-\\infty\\:<x<0,\\:x=0,\\:0<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$x^{4}+2x^{2}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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 $$x=0\\:$$ with domain:$${\\quad}x=0$$", "input": "x=0\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "x=0" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$0<x<\\infty\\:\\:$$ with domain:$${\\quad}0<x<\\infty\\:$$", "input": "0<x<\\infty\\:\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "0<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$-\\infty\\:<x<0\\:$$ with domain:$${\\quad}-\\infty\\:<x<0$$", "input": "-\\infty\\:<x<0\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "-\\infty\\:<x<0" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "step", "result": "-\\infty\\:<x<0,\\:x=0,\\:0<x<\\infty\\:" } ], "meta": { "interimType": "Combine Intervals With Domain 0Eq" } }, { "type": "step", "primary": "Summary of the monotone intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|}\\hline &-\\infty <x<0&x=0&0<x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime}}(x)<0&f {^{\\prime}}(x)=0&f {^{\\prime}}(x)>0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Decreasing}&\\mathrm{Minimum}&\\mathrm{Increasing}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Decreasing}:-\\infty\\:<x<0,\\:\\mathrm{Increasing}:0<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQnSGLMy6w719XZab5nwoBBIv9xEfQ6j65yOG5CkmPs8Cjeh7+jKEzLb7VNCEMF3Z/IP2sU6VWAmHt2BpuqN0ncCczd+yAO/2gwfvZJ8nHY18=" } }, { "type": "interim", "title": "Plug $$x=0\\:$$into $$x^{4}+2x^{2}:{\\quad}0$$", "input": "0^{4}+2\\cdot\\:0^{2}", "result": "\\mathrm{Minimum}\\left(0,\\:0\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "0" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7c/sJvxxypcyM4MeD1d2oSmOaXJQdDeoMwL6Na8beydAlN5Aod6Hr1Lp2e/29KhSgUy4GsClX2Z9vau7GMnhJzcI6qd66xVCZgYQjn+u3wikBDsS8Dph52h5Ln+bh6j/x7" } } ], "meta": { "solvingClass": "Function Extreme" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{4}+2x^{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }