critical x^4-x^2

{ "query": { "display": "critical points $$x^{4}-x^{2}$$", "symbolab_question": "FUNCTION#critical x^{4}-x^{2}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "critical", "default": "x=-\\frac{1}{\\sqrt{2}},x=0,x=\\frac{1}{\\sqrt{2}}" }, "steps": { "type": "interim", "title": "Critical Points of $$x^{4}-x^{2}:{\\quad}x=-\\frac{1}{\\sqrt{2}},\\:x=0,\\:x=\\frac{1}{\\sqrt{2}}$$", "input": "x^{4}-x^{2}", "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": "x^{4}-x^{2}", "result": "x=-\\frac{1}{\\sqrt{2}},\\:x=0,\\:x=\\frac{1}{\\sqrt{2}}", "steps": [ { "type": "interim", "title": "$$f^{\\prime}\\left(x\\right)=4x^{3}-2x$$", "input": "\\frac{d}{dx}\\left(x^{4}-x^{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(x^{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": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "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": "step", "result": "=4x^{3}-2x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Solve $$4x^{3}-2x=0:{\\quad}x=0,\\:x=-\\frac{1}{\\sqrt{2}},\\:x=\\frac{1}{\\sqrt{2}}$$", "input": "4x^{3}-2x=0", "steps": [ { "type": "interim", "title": "Factor $$4x^{3}-2x:{\\quad}2x\\left(\\sqrt{2}x+1\\right)\\left(\\sqrt{2}x-1\\right)$$", "input": "4x^{3}-2x", "steps": [ { "type": "interim", "title": "Factor out common term $$2x:{\\quad}2x\\left(2x^{2}-1\\right)$$", "input": "4x^{3}-2x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{3}=x^{2}x$$" ], "result": "=4x^{2}x-2x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Rewrite $$4$$ as $$2\\cdot\\:2$$", "result": "=2\\cdot\\:2x^{2}x-2x" }, { "type": "step", "primary": "Factor out common term $$2x$$", "result": "=2x\\left(2x^{2}-1\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=2x\\left(2x^{2}-1\\right)" }, { "type": "interim", "title": "Factor $$2x^{2}-1:{\\quad}\\left(\\sqrt{2}x+1\\right)\\left(\\sqrt{2}x-1\\right)$$", "input": "2x^{2}-1", "steps": [ { "type": "interim", "title": "Rewrite $$2x^{2}-1$$ as $$\\left(\\sqrt{2}x\\right)^{2}-1^{2}$$", "input": "2x^{2}-1", "result": "=\\left(\\sqrt{2}x\\right)^{2}-1^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$2=\\left(\\sqrt{2}\\right)^{2}$$" ], "result": "=\\left(\\sqrt{2}\\right)^{2}x^{2}-1", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=\\left(\\sqrt{2}\\right)^{2}x^{2}-1^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$\\left(\\sqrt{2}\\right)^{2}x^{2}=\\left(\\sqrt{2}x\\right)^{2}$$" ], "result": "=\\left(\\sqrt{2}x\\right)^{2}-1^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(\\sqrt{2}x\\right)^{2}-1^{2}=\\left(\\sqrt{2}x+1\\right)\\left(\\sqrt{2}x-1\\right)$$" ], "result": "=\\left(\\sqrt{2}x+1\\right)\\left(\\sqrt{2}x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=2x\\left(\\sqrt{2}x+1\\right)\\left(\\sqrt{2}x-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "2x\\left(\\sqrt{2}x+1\\right)\\left(\\sqrt{2}x-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\\:\\sqrt{2}x+1=0\\lor\\:\\sqrt{2}x-1=0" }, { "type": "interim", "title": "Solve $$\\sqrt{2}x+1=0:{\\quad}x=-\\frac{1}{\\sqrt{2}}$$", "input": "\\sqrt{2}x+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sqrt{2}x+1=0", "result": "\\sqrt{2}x=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\sqrt{2}x+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{2}x=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$\\sqrt{2}$$", "input": "\\sqrt{2}x=-1", "result": "x=-\\frac{1}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{2}$$", "result": "\\frac{\\sqrt{2}x}{\\sqrt{2}}=\\frac{-1}{\\sqrt{2}}" }, { "type": "step", "primary": "Simplify", "result": "x=-\\frac{1}{\\sqrt{2}}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$\\sqrt{2}x-1=0:{\\quad}x=\\frac{1}{\\sqrt{2}}$$", "input": "\\sqrt{2}x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sqrt{2}x-1=0", "result": "\\sqrt{2}x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\sqrt{2}x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{2}x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"step", "primary": "Simplify", "result": "x=\\frac{1}{\\sqrt{2}}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "x=0,\\:x=-\\frac{1}{\\sqrt{2}},\\:x=\\frac{1}{\\sqrt{2}}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=-\\frac{1}{\\sqrt{2}},\\:x=0,\\:x=\\frac{1}{\\sqrt{2}}" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/x4xq2YDRufFWghOsVoHbQs5F+N/82d9gQExbRFw+dZ46O4jPrNVvYsR23FCfYY1xU2RgZdW697FcK7XZ6FFM3ZMskHVr1WUkkoRbopLNBoA6/0zgGGGhNcYTnIAFOctM516rCvntU4ho/UbC5FNDUWB0BOtCKjjO52aLG340IQm6jVF5LdUKmTRWYV+yps/gazxVTC2YiFM25udqOdyHyfkZq8lQPQ/cEZBDLIblagFz" } }, { "type": "step", "primary": "Identify critical points not in $$f\\left(x\\right)$$ domain" }, { "type": "interim", "title": "Domain of $$x^{4}-x^{2}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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": "step", "primary": "All critical points are in domain", "result": "x=-\\frac{1}{\\sqrt{2}},\\:x=0,\\:x=\\frac{1}{\\sqrt{2}}" } ], "meta": { "solvingClass": "Function Critical" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{4}-x^{2}" }, "showViewLarger": true } } }