inflection-x^4+12x^3-12x+13

{ "query": { "display": "inflection points $$-x^{4}+12x^{3}-12x+13$$", "symbolab_question": "FUNCTION#inflection -x^{4}+12x^{3}-12x+13" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inflection", "default": "(0,13),(6,1237)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inflection Points of $$-x^{4}+12x^{3}-12x+13:{\\quad}\\left(0,\\:13\\right),\\:\\left(6,\\:1237\\right)$$", "steps": [ { "type": "definition", "title": "Inflection points definition", "text": "An inflection point is a point on the graph at which the second derivative is equal to zero or undefined and changes sign.<br/>If $$f''\\left(x\\right)>0\\:$$then $$f\\left(x\\right)\\:$$concave upwards.<br/>If $$f''\\left(x\\right)<0\\:$$then $$f\\left(x\\right)\\:$$concave downwards." }, { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)=-12x^{2}+72x$$", "input": "\\frac{d^{2}}{dx^{2}}\\left(-x^{4}+12x^{3}-12x+13\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-x^{4}+12x^{3}-12x+13\\right)=-4x^{3}+36x^{2}-12$$", "input": "\\frac{d}{dx}\\left(-x^{4}+12x^{3}-12x+13\\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(12x^{3}\\right)-\\frac{d}{dx}\\left(12x\\right)+\\frac{d}{dx}\\left(13\\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": "$$\\frac{d}{dx}\\left(12x^{3}\\right)=36x^{2}$$", "input": "\\frac{d}{dx}\\left(12x^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\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": "=12\\cdot\\:3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=36x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuQb8RYXoFIwBbvw17sYq+QcjlLRK1jUV206qo4+vRN7tT4VHRMGmsVZGiCjQ1g6xodEeJB8NSwK2cnf2Bc1WE1PXEpfcfbu02Nx1nf7/MLiotg8LmC3GqUZVM1dfkCAjo8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(12x\\right)=12$$", "input": "\\frac{d}{dx}\\left(12x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=12\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=12", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhF+Ykp5nUgfRlgPAvwoz5KXIQHgliMhSOSNsNni19In94H8CoGnrS97E87MitDaQA4bfwiV6iMLJ5sC1nL7dOZiPtq11pJT4yOnp/GI6P1RsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(13\\right)=0$$", "input": "\\frac{d}{dx}\\left(13\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtxPJT6Jpa79CIl+OYj2rqXZGku9zFkxwe1dTH8vycb9TbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51gxCPg47aOih0Pc4rxS61eT" } }, { "type": "step", "result": "=-4x^{3}+36x^{2}-12+0" }, { "type": "step", "primary": "Simplify", "result": "=-4x^{3}+36x^{2}-12", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(-4x^{3}+36x^{2}-12\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-4x^{3}+36x^{2}-12\\right)=-12x^{2}+72x$$", "input": "\\frac{d}{dx}\\left(-4x^{3}+36x^{2}-12\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dx}\\left(4x^{3}\\right)+\\frac{d}{dx}\\left(36x^{2}\\right)-\\frac{d}{dx}\\left(12\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x^{3}\\right)=12x^{2}$$", "input": "\\frac{d}{dx}\\left(4x^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\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": "=4\\cdot\\:3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=12x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuytnrDSBQVECiAN7hMFdqKTdaV09PMxEKZ9FieghTFwbVQDmNnvMzBhKnFOUzUT515NkzKQgtswLlLi9MgL+gq5QV7agSZLIzF7D9vX0CHvx8XaWXbYXWiYPXxbVFoLirCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(36x^{2}\\right)=72x$$", "input": "\\frac{d}{dx}\\left(36x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=36\\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": "=36\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=72x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqxyW/0wKf8REBsy0UImeSIcjlLRK1jUV206qo4+vRN7meShn5FwCcvCqfLkfi7dJz/L0MoYg+CUn6oyL3EO7YqXR80qtCHTdqFPuotQcIdrwr+VItrLVy4h613FJ4GLtA==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(12\\right)=0$$", "input": "\\frac{d}{dx}\\left(12\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiV8QO16oD4N5U13Ecg0LSbZGku9zFkxwe1dTH8vycb9TbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51hXMiCELLqHZHbZXScipdva" } }, { "type": "step", "result": "=-12x^{2}+72x-0" }, { "type": "step", "primary": "Simplify", "result": "=-12x^{2}+72x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-12x^{2}+72x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Concave Downward$$:-\\infty\\:<x<0,\\:$$Concave Upward$$:0<x<6,\\:$$Concave Downward$$:6<x<\\infty\\:$$", "input": "f\\:{^{\\prime\\prime}}\\left(x\\right)=-12x^{2}+72x", "steps": [ { "type": "interim", "title": "Find where $$f\\:{^{\\prime\\prime}}\\left(x\\right)$$ is equal to zero or undefined:$${\\quad}x=0,\\:x=6$$", "steps": [ { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)=0:{\\quad}x=0,\\:x=6$$", "input": "-12x^{2}+72x=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-12x^{2}+72x=0", "result": "{x}_{1,\\:2}=\\frac{-72\\pm\\:\\sqrt{72^{2}-4\\left(-12\\right)\\cdot\\:0}}{2\\left(-12\\right)}", "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 $${\\quad}a=-12,\\:b=72,\\:c=0$$", "result": "{x}_{1,\\:2}=\\frac{-72\\pm\\:\\sqrt{72^{2}-4\\left(-12\\right)\\cdot\\:0}}{2\\left(-12\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{72^{2}-4\\left(-12\\right)\\cdot\\:0}=72$$", "input": "\\sqrt{72^{2}-4\\left(-12\\right)\\cdot\\:0}", "result": "{x}_{1,\\:2}=\\frac{-72\\pm\\:72}{2\\left(-12\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{72^{2}+4\\cdot\\:12\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=\\sqrt{72^{2}+0}" }, { "type": "step", "primary": "$$72^{2}+0=72^{2}$$", "result": "=\\sqrt{72^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=72", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kP+2kPknikJKE9DE9AhIgRcTbxdr5Ie6BPRBOowFyCktOtZYwUjyXhDTsNnn6ElrveAEknkw3iwvN9zzlnv78qN6Hv6MoTMtvtU0IQwXdn9Yzwt0R4NGK72IiC9mz86v9zRFMZd7BJWt2U/zprgIESS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-72+72}{2\\left(-12\\right)},\\:{x}_{2}=\\frac{-72-72}{2\\left(-12\\right)}" }, { "type": "interim", "title": "$$x=\\frac{-72+72}{2\\left(-12\\right)}:{\\quad}0$$", "input": "\\frac{-72+72}{2\\left(-12\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-72+72}{-2\\cdot\\:12}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-72+72=0$$", "result": "=\\frac{0}{-2\\cdot\\:12}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:12=24$$", "result": "=\\frac{0}{-24}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{0}{24}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jmn27RQSwipRUH3QDI0UYu/a3SRFx74mMT+HESD/brp1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl73Tcan7wFkSuOUEIM4ZHqHELNklisX3wkw8pIfT5pkvJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$x=\\frac{-72-72}{2\\left(-12\\right)}:{\\quad}6$$", "input": "\\frac{-72-72}{2\\left(-12\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-72-72}{-2\\cdot\\:12}" }, { "type": "step", "primary": "Subtract the numbers: $$-72-72=-144$$", "result": "=\\frac{-144}{-2\\cdot\\:12}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:12=24$$", "result": "=\\frac{-144}{-24}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{144}{24}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{144}{24}=6$$", "result": "=6" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mP52iauJTVrvXRi5A2Iip+/a3SRFx74mMT+HESD/brp1g99dC9fj9sg0EHzBIRDRjBGJGUdgcV1QE8NtKLFEFb3Tcan7wFkSuOUEIM4ZHqG+97Ch90yeFhDXlK2RoXt9JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=0,\\:x=6" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=0,\\:x=6" } ], "meta": { "interimType": "Explore Function Slope Zero Specific 1Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)>0:{\\quad}0<x<6$$", "input": "-12x^{2}+72x>0", "steps": [ { "type": "interim", "title": "Rewrite in standard form", "input": "-12x^{2}+72x>0", "result": "-x^{2}+6x>0", "steps": [ { "type": "step", "primary": "Divide both sides by $$12$$", "result": "-\\frac{12x^{2}}{12}+\\frac{72x}{12}>\\frac{0}{12}" }, { "type": "step", "primary": "Simplify", "result": "-x^{2}+6x>0", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$-x^{2}+6x:{\\quad}-x\\left(x-6\\right)$$", "input": "-x^{2}+6x", "result": "-x\\left(x-6\\right)>0", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=-xx+6x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$-x$$", "result": "=-x\\left(x-6\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\left(-x\\left(x-6\\right)\\right)\\left(-1\\right)<0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "x\\left(x-6\\right)<0", "meta": { "solvingClass": "Solver" } }, { "type": "interim", "title": "Identify the intervals", "result": "0<x<6", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$x\\left(x-6\\right)$$" }, { "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 $$x-6$$", "steps": [ { "type": "interim", "title": "$$x-6=0:{\\quad}x=6$$", "input": "x-6=0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "x-6=0", "result": "x=6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "x-6+6=0+6" }, { "type": "step", "primary": "Simplify", "result": "x=6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$-x^{2}+6x:{\\quad}-x\\left(x-6\\right)$$", "input": "-x^{2}+6x", "result": "-x\\left(x-6\\right)<0", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=-xx+6x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$-x$$", "result": "=-x\\left(x-6\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Multiply both sides by $$-1$$ (reverse the inequality)", "result": "\\left(-x\\left(x-6\\right)\\right)\\left(-1\\right)>0\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": 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{ "interimType": "Find Sign 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|}\\hline &x<0&x=0&0<x<6&x=6&x>6\\\\\\hline x&-&0&+&+&+\\\\\\hline x-6&-&-&-&0&+\\\\\\hline x(x-6)&+&0&-&0&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$>\\:0$$", "result": "x<0\\lor\\:x>6" } ], "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<6,\\:x=6,\\:6<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$-x^{4}+12x^{3}-12x+13\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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": 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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 $$x=6\\:$$ with domain:$${\\quad}x=6$$", "input": "x=6\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "x=6" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "interim", "title": "Combine $$0<x<6\\:$$ with domain:$${\\quad}0<x<6$$", "input": "0<x<6\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "0<x<6" } ], "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": "interim", "title": "Combine $$6<x<\\infty\\:\\:$$ with domain:$${\\quad}6<x<\\infty\\:$$", "input": "6<x<\\infty\\:\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "6<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "step", "result": "-\\infty\\:<x<0,\\:x=0,\\:0<x<6,\\:x=6,\\:6<x<\\infty\\:" } ], "meta": { "interimType": "Combine Intervals With Domain 0Eq" } }, { "type": "step", "primary": "Summary of the sign intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|}\\hline &-\\infty <x<0&x=0&0<x<6&x=6&6<x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime\\prime}}(x)<0&f {^{\\prime\\prime}}(x)=0&f {^{\\prime\\prime}}(x)>0&f {^{\\prime\\prime}}(x)=0&f {^{\\prime\\prime}}(x)<0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Concave\\:Downward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Upward}&\\mathrm{Inflection}&\\mathrm{Concave\\:Downward}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Concave\\:Downward}:-\\infty\\:<x<0,\\:\\mathrm{Concave\\:Upward}:0<x<6,\\:\\mathrm{Concave\\:Downward}:6<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQWS/fnqlAfkusYvPtw7A/trZ0LNZGIpJVMJCy6nxlCyvmLxM0G9CBRMGh6bvrcjLEeqXxdc+rps1CUyb7fqI2GTNzkA+TOIh4/25jQMNv0DmwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Plug $$x=0\\:$$into $$-x^{4}+12x^{3}-12x+13:{\\quad}13$$", "input": "-0^{4}+12\\cdot\\:0^{3}-12\\cdot\\:0+13", "result": "\\left(0,\\:13\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "13" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7cNsGOPam4zCDF6zitJdcFOiR1hevuq25FXRIClbdNx1TnLn4bn3h1frswkVh6bktzC10n7d918LP8zfQ2JqqHSADVW6tatq215Cpr+r11ZfL4qYhxXKy6IFNvKo41DGN3UZ3PiZtsg2Yf7mQl+mhO5w==" } }, { "type": "interim", "title": "Plug $$x=6\\:$$into $$-x^{4}+12x^{3}-12x+13:{\\quad}1237$$", "input": "-6^{4}+12\\cdot\\:6^{3}-12\\cdot\\:6+13", "result": "\\left(6,\\:1237\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "1237" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7c9zMcz3UdzmGWatXsbSPDuiR1hevuq25FXRIClbdNx1TnLn4bn3h1frswkVh6bktzC10n7d918LP8zfQ2JqqHSDIV0bMuERWXTtOtYrey/G/4qYhxXKy6IFNvKo41DGN3UZ3PiZtsg2Yf7mQl+mhO5w==" } }, { "type": "step", "result": "\\left(0,\\:13\\right),\\:\\left(6,\\:1237\\right)" } ], "meta": { "solvingClass": "Function Inflection" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-x^{4}+12x^{3}-12x+13" }, "showViewLarger": true } }, "meta": { "showVerify": true } }