extreme f(x)=4x^3-48

{ "query": { "display": "extreme points $$f\\left(x\\right)=4x^{3}-48$$", "symbolab_question": "FUNCTION#extreme f(x)=4x^{3}-48" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "extreme", "default": "\\mathrm{Saddle}(0,-48)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find using the First Derivative Test", "query": { "display": "first derivative test $$4x^{3}-48$$", "symbolab_question": "firstderivativetest 4x^{3}-48" } }, { "method": "Find using the Second Derivative Test", "query": { "display": "second derivative test $$4x^{3}-48$$", "symbolab_question": "secondderivativetest 4x^{3}-48" } } ], "steps": { "type": "interim", "title": "Extreme Points of $$4x^{3}-48:{\\quad}$$Saddle$$\\left(0,\\:-48\\right)$$", "steps": [ { "type": "definition", "title": "First Derivative Test definition", "text": "Suppose that $$x=c$$ is a critical point of $$f\\left(x\\right)$$ then, <br/>If $$f\\:{^{\\prime}}\\left(x\\right)>0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the right of $$x=c$$ then $$x=c$$ is a local maximum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)>\\:0$$ to the right of $$x=c$$ then $$x=c$$ is a local minimum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)$$ is the same sign on both sides of $$x=c$$ then $$x=c$$ is neither a local maximum nor a local minimum." }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)=12x^{2}$$", "input": "\\frac{d}{dx}\\left(4x^{3}-48\\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(48\\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": "$$\\frac{d}{dx}\\left(48\\right)=0$$", "input": "\\frac{d}{dx}\\left(48\\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+idP5vLVrjUll6eMdYgHQjaj3OvP5n/PPAd7+ULLZGku9zFkxwe1dTH8vycb9TbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51jqH16ctd2wF8vSWr1qWRKL" } }, { "type": "step", "result": "=12x^{2}-0" }, { "type": "step", "primary": "Simplify", "result": "=12x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Increasing$$:-\\infty\\:<x<0,\\:$$Increasing$$:0<x<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(x\\right)=12x^{2}", "steps": [ { "type": "interim", "title": "Find the critical points:$${\\quad}x=0$$", "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": "$$f\\:{^{\\prime}}\\left(x\\right)=0:{\\quad}x=0$$", "input": "12x^{2}=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$12$$", "input": "12x^{2}=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$12$$", "input": "12x^{2}=0", "result": "x^{2}=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$12$$", "result": "\\frac{12x^{2}}{12}=\\frac{0}{12}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq" } }, { "type": "step", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=0" } ], "meta": { "interimType": "Explore Function Slope Zero Title 0Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)>0:{\\quad}x<0\\lor\\:x>0$$", "input": "12x^{2}>0", "steps": [ { "type": "interim", "title": "Divide both sides by $$12$$", "input": "12x^{2}>0", "result": "x^{2}>0", "steps": [ { "type": "step", "primary": "Divide both sides by $$12$$", "result": "\\frac{12x^{2}}{12}>\\frac{0}{12}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}>0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$u^n\\:>\\:0$$, if $$n\\:$$is even then $$u\\:<\\:0\\quad\\:$$or$$\\quad\\:\\:u\\:>\\:0$$" }, { "type": "step", "result": "x<0\\lor\\:x>0" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(x\\right)<0:{\\quad}$$False for all $$x\\in\\mathbb{R}$$", "input": "12x^{2}<0", "steps": [ { "type": "interim", "title": "Divide both sides by $$12$$", "input": "12x^{2}<0", "result": "x^{2}<0", "steps": [ { "type": "step", "primary": "Divide both sides by $$12$$", "result": "\\frac{12x^{2}}{12}<\\frac{0}{12}" }, { "type": "step", "primary": "Simplify", "result": "x^{2}<0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "If n is even, $$u^n\\:\\ge\\:0\\:$$ for all $$u$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "-\\infty\\:<x<0,\\:x=0,\\:0<x<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of $$4x^{3}-48\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "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": "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 $$-\\infty\\:<x<0\\:$$ 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 $$0<x<\\infty\\:\\:$$ with domain:$${\\quad}0<x<\\infty\\:$$", "input": "0<x<\\infty\\:\\land\\:\\mathrm{True\\:for\\:all}\\:x\\in\\mathbb{R}", "steps": [ { "type": "step", "primary": "Simplify", "result": "0<x<\\infty\\:" } ], "meta": { "interimType": "Combine With Domain 1Eq" } }, { "type": "step", "result": "-\\infty\\:<x<0,\\:x=0,\\:0<x<\\infty\\:" } ], "meta": { "interimType": "Combine Intervals With Domain 0Eq" } }, { "type": "step", "primary": "Summary of the monotone intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|}\\hline &-\\infty <x<0&x=0&0<x<\\infty \\\\\\hline \\mathrm{Sign}&f {^{\\prime}}(x)>0&f {^{\\prime}}(x)=0&f {^{\\prime}}(x)>0\\\\\\hline \\mathrm{Behavior}&\\mathrm{Increasing}&\\mathrm{Saddle}&\\mathrm{Increasing}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Increasing}:-\\infty\\:<x<0,\\:\\mathrm{Increasing}:0<x<\\infty\\:" } ], "meta": { "interimType": "Function Find Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGUL8PhNw/UnccDjDvkOKjBsievSRG8FbtJ67MZcqchnBIfFLvhIw/699NIWZ6gDQnSGLMy6w719XZab5nwoBBJYei62hJyo5+JPDusOdE9LvbBmbuQNTF0TphKZ8RuvaF3/5bmG/m8HhSU4CydsVjf8Ceubne4/cZFNlCz6BtQc=" } }, { "type": "interim", "title": "Plug $$x=0\\:$$into $$4x^{3}-48:{\\quad}-48$$", "input": "4\\cdot\\:0^{3}-48", "result": "\\mathrm{Saddle}\\left(0,\\:-48\\right)", "steps": [ { "type": "step", "primary": "Simplify", "result": "-48" } ], "meta": { "interimType": "Generic Plug Into Specific 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3d3CfN4XlvQgt3kPeYLw7cFDODllvwaN2OY9t7hXMjDZsIScO8r/AesOln5vT0RwjQ/EUarKSmNhAi0MiN0Ss1byjazmkHKLeXVFv0BFQJ4XBeNZiO50+6w6MGCoCFZNG/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "solvingClass": "Function Extreme" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "4x^{3}-48" }, "showViewLarger": true } }, "meta": { "showVerify": true } }