critical-5x^4-x^3+2x^2

{ "query": { "display": "critical points $$-5x^{4}-x^{3}+2x^{2}$$", "symbolab_question": "FUNCTION#critical -5x^{4}-x^{3}+2x^{2}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "critical", "default": "x=\\frac{-3-\\sqrt{329}}{40},x=0,x=\\frac{-3+\\sqrt{329}}{40}" }, "steps": { "type": "interim", "title": "Critical Points of $$-5x^{4}-x^{3}+2x^{2}:{\\quad}x=\\frac{-3-\\sqrt{329}}{40},\\:x=0,\\:x=\\frac{-3+\\sqrt{329}}{40}$$", "input": "-5x^{4}-x^{3}+2x^{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": "-5x^{4}-x^{3}+2x^{2}", "result": "x=\\frac{-3-\\sqrt{329}}{40},\\:x=0,\\:x=\\frac{-3+\\sqrt{329}}{40}", "steps": [ { "type": "interim", "title": "$$f^{\\prime}\\left(x\\right)=-20x^{3}-3x^{2}+4x$$", "input": "\\frac{d}{dx}\\left(-5x^{4}-x^{3}+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(5x^{4}\\right)-\\frac{d}{dx}\\left(x^{3}\\right)+\\frac{d}{dx}\\left(2x^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5x^{4}\\right)=20x^{3}$$", "input": "\\frac{d}{dx}\\left(5x^{4}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=5\\frac{d}{dx}\\left(x^{4}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=5\\cdot\\:4x^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=20x^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuaIWboyJjcmGa7VgP571ZaTdaV09PMxEKZ9FieghTFwmvxRL4hmKP9Y2xjGkQoL9WYYnHLVfzWgXhHqK+R1GFq5QV7agSZLIzF7D9vX0CHvHaMJZNAfxRwFkO2+yrS2/bCI2sSeA74029n2yo277ZU=" } }, { "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": "$$\\frac{d}{dx}\\left(2x^{2}\\right)=4x$$", "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": "=-20x^{3}-3x^{2}+4x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Solve $$-20x^{3}-3x^{2}+4x=0:{\\quad}x=0,\\:x=\\frac{-3+\\sqrt{329}}{40},\\:x=\\frac{-3-\\sqrt{329}}{40}$$", "input": "-20x^{3}-3x^{2}+4x=0", "steps": [ { "type": "interim", "title": "Factor $$-20x^{3}-3x^{2}+4x:{\\quad}-x\\left(20x^{2}+3x-4\\right)$$", "input": "-20x^{3}-3x^{2}+4x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$", "$$x^{3}=x^{2}x$$" ], "result": "=-20x^{2}x-3xx+4x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$-x$$", "result": "=-x\\left(20x^{2}+3x-4\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "-x\\left(20x^{2}+3x-4\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "x=0\\lor\\:20x^{2}+3x-4=0" }, { "type": "interim", "title": "Solve $$20x^{2}+3x-4=0:{\\quad}x=\\frac{-3+\\sqrt{329}}{40},\\:x=\\frac{-3-\\sqrt{329}}{40}$$", "input": "20x^{2}+3x-4=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "20x^{2}+3x-4=0", "result": "{x}_{1,\\:2}=\\frac{-3\\pm\\:\\sqrt{3^{2}-4\\cdot\\:20\\left(-4\\right)}}{2\\cdot\\:20}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=20,\\:b=3,\\:c=-4$$", "result": "{x}_{1,\\:2}=\\frac{-3\\pm\\:\\sqrt{3^{2}-4\\cdot\\:20\\left(-4\\right)}}{2\\cdot\\:20}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{3^{2}-4\\cdot\\:20\\left(-4\\right)}=\\sqrt{329}$$", "input": "\\sqrt{3^{2}-4\\cdot\\:20\\left(-4\\right)}", "result": "{x}_{1,\\:2}=\\frac{-3\\pm\\:\\sqrt{329}}{2\\cdot\\:20}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{3^{2}+4\\cdot\\:20\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:20\\cdot\\:4=320$$", "result": "=\\sqrt{3^{2}+320}" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\sqrt{9+320}" }, { "type": "step", "primary": "Add the numbers: $$9+320=329$$", "result": "=\\sqrt{329}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I8nbFv5ohMEv/aq9F0opf23qWKuSUObp3I5HiyPEJpVV00rpv8+ZC6TM10tVCSHsoDcnz6I6EbFIZGDoyqUAx5ym7jwffWC2+rVLogr4bjAwfnTaFQARzoIip8gZfUe5RTbdKbqftCV6NLKnVxnviwC3HmKRZJXWqmS1qw7DP7ckt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-3+\\sqrt{329}}{2\\cdot\\:20},\\:{x}_{2}=\\frac{-3-\\sqrt{329}}{2\\cdot\\:20}" }, { "type": "interim", "title": "$$x=\\frac{-3+\\sqrt{329}}{2\\cdot\\:20}:{\\quad}\\frac{-3+\\sqrt{329}}{40}$$", "input": "\\frac{-3+\\sqrt{329}}{2\\cdot\\:20}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:20=40$$", "result": "=\\frac{-3+\\sqrt{329}}{40}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7S06bDB+MJbrOisU4Z6oFiPrPr56hX/hyqeC/mCQ+Z3/GvPSRcA2I0vLd3rOOIoYLzMFYmi1F5Hg/ibpEToVnY47XFRivRbGxBrjjS1604EFYa3z3g47AsvY0DWUM9WZvFg2QliWVtdiykO19PgU04ugF6XXBFJVDwyH+r/dxdOdNyFzAENZzyb6r73kF0RG1vzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$x=\\frac{-3-\\sqrt{329}}{2\\cdot\\:20}:{\\quad}\\frac{-3-\\sqrt{329}}{40}$$", "input": "\\frac{-3-\\sqrt{329}}{2\\cdot\\:20}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:20=40$$", "result": "=\\frac{-3-\\sqrt{329}}{40}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QCrI66+7EgSAvN8Xk/GTa/rPr56hX/hyqeC/mCQ+Z3/GvPSRcA2I0vLd3rOOIoYLzMFYmi1F5Hg/ibpEToVnY2jwQPRX2uAFlSqX37lE/cxYa3z3g47AsvY0DWUM9WZvFg2QliWVtdiykO19PgU04s/Yz1uFeesIaVuTylHIyGZNyFzAENZzyb6r73kF0RG1vzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=\\frac{-3+\\sqrt{329}}{40},\\:x=\\frac{-3-\\sqrt{329}}{40}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "x=0,\\:x=\\frac{-3+\\sqrt{329}}{40},\\:x=\\frac{-3-\\sqrt{329}}{40}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{-3-\\sqrt{329}}{40},\\:x=0,\\:x=\\frac{-3+\\sqrt{329}}{40}" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/x3l5erYtKTbOfqVrkQIvh85/dr08XsHmtWtDmXL7M/rr7WQg0UsQt1aSg4OK5ATVfCyEl/RW3NF9wWEPfaKFEE8myshBylEgFF4wsaY+xS5tVbHT1tAXn5Bq9/WY6M4/IqN6Hv6MoTMtvtU0IQwXdn/tcZ/ECY+sBUB+A6V/carltrVtPXzKGAb2BNfFdo/SbhMgHIwbQPRHPbacB7yleDMskHVr1WUkkoRbopLNBoA6i1D0ijqMBRGRlOwoJTKotg==" } }, { "type": "step", "primary": "Identify critical points not in $$f\\left(x\\right)$$ domain" }, { "type": "interim", "title": "Domain of $$-5x^{4}-x^{3}+2x^{2}\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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": "step", "primary": "All critical points are in domain", "result": "x=\\frac{-3-\\sqrt{329}}{40},\\:x=0,\\:x=\\frac{-3+\\sqrt{329}}{40}" } ], "meta": { "solvingClass": "Function Critical" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-5x^{4}-x^{3}+2x^{2}" }, "showViewLarger": true } } }