critical x^4-2x^3+x^2

{ "query": { "display": "critical points $$x^{4}-2x^{3}+x^{2}$$", "symbolab_question": "FUNCTION#critical x^{4}-2x^{3}+x^{2}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "critical", "default": "x=0,x=\\frac{1}{2},x=1" }, "steps": { "type": "interim", "title": "Critical Points of $$x^{4}-2x^{3}+x^{2}:{\\quad}x=0,\\:x=\\frac{1}{2},\\:x=1$$", "input": "x^{4}-2x^{3}+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}-2x^{3}+x^{2}", "result": "x=0,\\:x=\\frac{1}{2},\\:x=1", "steps": [ { "type": "interim", "title": "$$f^{\\prime}\\left(x\\right)=4x^{3}-6x^{2}+2x$$", "input": "\\frac{d}{dx}\\left(x^{4}-2x^{3}+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(2x^{3}\\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": "$$\\frac{d}{dx}\\left(2x^{3}\\right)=6x^{2}$$", "input": "\\frac{d}{dx}\\left(2x^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(x^{3}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2\\cdot\\:3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnw/L9p+IMyouNhbzU7bNo6TdaV09PMxEKZ9FieghTFwWn+xgzF413/kfGYgWrraBxHO0oTnnZveyzJ4AtC1ZGNjDT5Dj/fM73/u0bafjbUvQ9geJFbKg/ol2YLcmIH9YSS3daIZHtloJpe/PvtsyNI=" } }, { "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}-6x^{2}+2x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Solve $$4x^{3}-6x^{2}+2x=0:{\\quad}x=0,\\:x=\\frac{1}{2},\\:x=1$$", "input": "4x^{3}-6x^{2}+2x=0", "steps": [ { "type": "interim", "title": "Factor $$4x^{3}-6x^{2}+2x:{\\quad}2x\\left(2x-1\\right)\\left(x-1\\right)$$", "input": "4x^{3}-6x^{2}+2x", "steps": [ { "type": "interim", "title": "Factor out common term $$2x:{\\quad}2x\\left(2x^{2}-3x+1\\right)$$", "input": "4x^{3}-6x^{2}+2x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$", "$$x^{3}=x^{2}x$$" ], "result": "=4x^{2}x-6xx+2x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Rewrite $$6$$ as $$2\\cdot\\:3$$", "secondary": [ "Rewrite $$4$$ as $$2\\cdot\\:2$$" ], "result": "=2\\cdot\\:2x^{2}x-2\\cdot\\:3xx+2x" }, { "type": "step", "primary": "Factor out common term $$2x$$", "result": "=2x\\left(2x^{2}-3x+1\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=2x\\left(2x^{2}-3x+1\\right)" }, { "type": "interim", "title": "Factor $$2x^{2}-3x+1:{\\quad}\\left(2x-1\\right)\\left(x-1\\right)$$", "input": "2x^{2}-3x+1", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "2x^{2}-3x+1", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=2,\\:b=-3,\\:c=1$$", "$$u*v=2,\\:u+v=-3$$" ] }, { "type": "interim", "title": "Factors of $$2:{\\quad}1,\\:2$$", "input": "2", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRpp3lFwBgr08J1hDIhHaqLjwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJmyKVlUUECPEUZsSDkCFf8u/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$2$$", "result": "1,\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$2:{\\quad}-1,\\:-2$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=2,\\:$$check if $$u+v=-3$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=2:\\quad\\:u*v=2,\\:u+v=3\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=-1,\\:v=-2:\\quad\\:u*v=2,\\:u+v=-3\\quad\\Rightarrow\\quad\\:$$True" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-1,\\:v=-2" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(2x^{2}-x\\right)+\\left(-2x+1\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwgGxSK0omOx07ysOBVs9Ia4rnvpYQ0aygJfymuicdqpZxbXdJC0VO14XGaS2e1N17dyfY+M+XdwfX2DisUuAkI+MP3J2SPIBsjnqRSuqAPTw+81tdhJmYEhYwvkzI+pEW5rqMyF3nXkDK7FLxfFx124Zi9wpyEKv79aqbxM02VJiJ4DPJZQ/Gd4qNOAL0ITrMb8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "result": "=\\left(2x^{2}-x\\right)+\\left(-2x+1\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$2x^{2}-x:\\quad\\:x\\left(2x-1\\right)$$", "input": "2x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=2xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(2x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-1\\:$$from $$-2x+1:\\quad\\:-\\left(2x-1\\right)$$", "input": "-2x+1", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(2x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tEQLOeLR1SMzmbFdGNRoJZN1pXT08zEQpn0WJ6CFMXD9pNnvGSE4c+RSRHJvBJPw0TgtQnan6YKJOTynJ7KWXyGk8vIJisuT2N3pfkW1JpawooBe7OKH4/NA59T9vDuxuhvtrPBMHJcjkx5jJinPkdXLZDQn5msW4WKW+lq/Hgs=" } }, { "type": "step", "result": "=x\\left(2x-1\\right)-\\left(2x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$2x-1$$", "result": "=\\left(2x-1\\right)\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=2x\\left(2x-1\\right)\\left(x-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "2x\\left(2x-1\\right)\\left(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\\:2x-1=0\\lor\\:x-1=0" }, { "type": "interim", "title": "Solve $$2x-1=0:{\\quad}x=\\frac{1}{2}$$", "input": "2x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2x-1=0", "result": "2x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "2x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "2x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=1", "result": "x=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{1}{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 $$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Move to the Right Title 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}{2},\\:x=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=0,\\:x=\\frac{1}{2},\\:x=1" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/x4xq2YDRufFWghOsVoHbQs6HAyZuN7DVUuMf+rAiGMjnIU26QE5P12wzg18qG9/95qFc+xUlM6NC9tbIYWNj5j48VUwtmIhTNubnajnch8n5/Pr23TX8AmqPbo+Hv0+ek+9sGZu5A1MXROmEpnxG69onrTLra3zck+M4W3iIDthkmC/+JA12tfS7dbgjr8XIPESMo62igsObuwlZQktw8Q28btwYXu7RnF2bfMUQyo0DlqUU3/t5gHd6wc9P2qVR1w==" } }, { "type": "step", "primary": "Identify critical points not in $$f\\left(x\\right)$$ domain" }, { "type": "interim", "title": "Domain of $$x^{4}-2x^{3}+x^{2}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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=0,\\:x=\\frac{1}{2},\\:x=1" } ], "meta": { "solvingClass": "Function Critical" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{4}-2x^{3}+x^{2}" }, "showViewLarger": true } } }