inflection f(x)=x^3-3x^2+4

{ "query": { "display": "inflection points $$f\\left(x\\right)=x^{3}-3x^{2}+4$$", "symbolab_question": "FUNCTION#inflection f(x)=x^{3}-3x^{2}+4" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inflection", "default": "(1,2)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inflection Points of $$x^{3}-3x^{2}+4:{\\quad}\\left(1,\\:2\\right)$$", "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": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)=6x-6$$", "input": "\\frac{d^{2}}{dx^{2}}\\left(x^{3}-3x^{2}+4\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{3}-3x^{2}+4\\right)=3x^{2}-6x$$", "input": "\\frac{d}{dx}\\left(x^{3}-3x^{2}+4\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{3}\\right)-\\frac{d}{dx}\\left(3x^{2}\\right)+\\frac{d}{dx}\\left(4\\right)" }, { "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": "$$\\frac{d}{dx}\\left(3x^{2}\\right)=6x$$", "input": "\\frac{d}{dx}\\left(3x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\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": "=3\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoMCvILd9ATo1x5jS/pE9/WTdaV09PMxEKZ9FieghTFwhVeofZ6TN6CjAjiGv3Kp4KN6Hv6MoTMtvtU0IQwXdn+0Pe4MM1RA6/7KiCIasGeudcfHQ5Gx5B84odmIQ8SsiA==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\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+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "type": "step", "result": "=3x^{2}-6x+0" }, { "type": "step", "primary": "Simplify", "result": "=3x^{2}-6x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(3x^{2}-6x\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x^{2}-6x\\right)=6x-6$$", "input": "\\frac{d}{dx}\\left(3x^{2}-6x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(3x^{2}\\right)-\\frac{d}{dx}\\left(6x\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x^{2}\\right)=6x$$", "input": "\\frac{d}{dx}\\left(3x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\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": "=3\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoMCvILd9ATo1x5jS/pE9/WTdaV09PMxEKZ9FieghTFwhVeofZ6TN6CjAjiGv3Kp4KN6Hv6MoTMtvtU0IQwXdn+0Pe4MM1RA6/7KiCIasGeudcfHQ5Gx5B84odmIQ8SsiA==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(6x\\right)=6$$", "input": "\\frac{d}{dx}\\left(6x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=6\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=6", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhppyBxBdApfk+SBOUguWFnZGku9zFkxwe1dTH8vycb9iU7QbEXaQW7Sg4nSYYaTV1NbbqpyK7JQEZdATEJR51hjHS9CfrzgF/gRWKQsm38l" } }, { "type": "step", "result": "=6x-6" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=6x-6" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Concave Downward$$:-\\infty\\:<x<1,\\:$$Concave Upward$$:1<x<\\infty\\:$$", "input": "f\\:{^{\\prime\\prime}}\\left(x\\right)=6x-6", "steps": [ { "type": "interim", "title": "Find where $$f\\:{^{\\prime\\prime}}\\left(x\\right)$$ is equal to zero or undefined:$${\\quad}x=1$$", "steps": [ { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)=0:{\\quad}x=1$$", "input": "6x-6=0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "6x-6=0", "result": "6x=6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "6x-6+6=0+6" }, { "type": "step", "primary": "Simplify", "result": "6x=6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x=6", "result": "x=1", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x}{6}=\\frac{6}{6}" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=1" } ], "meta": { "interimType": "Explore Function Slope Zero Specific 1Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)>0:{\\quad}x>1$$", "input": "6x-6>0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "6x-6>0", "result": "6x>6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "6x-6+6>0+6" }, { "type": "step", "primary": "Simplify", "result": "6x>6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x>6", "result": "x>1", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x}{6}>\\frac{6}{6}" }, { "type": "step", "primary": "Simplify", "result": "x>1" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime\\prime}}\\left(x\\right)<0:{\\quad}x<1$$", "input": "6x-6<0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "6x-6<0", "result": "6x<6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "6x-6+6<0+6" }, { "type": "step", "primary": "Simplify", "result": "6x<6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. 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