extreme f(x)=-0.1t^2+0.8t+98.8

{ "query": { "display": "extreme points $$f\\left(x\\right)=-0.1t^{2}+0.8t+98.8$$", "symbolab_question": "FUNCTION#extreme f(x)=-0.1t^{2}+0.8t+98.8" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "extreme", "default": "\\mathrm{Maximum}(4,100.4)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find using the First Derivative Test", "query": { "display": "first derivative test $$-0.1t^{2}+0.8t+98.8$$", "symbolab_question": "firstderivativetest -0.1t^{2}+0.8t+98.8" } }, { "method": "Find using the Second Derivative Test", "query": { "display": "second derivative test $$-0.1t^{2}+0.8t+98.8$$", "symbolab_question": "secondderivativetest -0.1t^{2}+0.8t+98.8" } } ], "steps": { "type": "interim", "title": "Extreme Points of $$-0.1t^{2}+0.8t+98.8:{\\quad}$$Maximum$$\\left(4,\\:100.4\\right)$$", "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": "$$f\\:{^{\\prime}}\\left(t\\right)=-0.2t+0.8$$", "input": "\\frac{d}{dt}\\left(-0.1t^{2}+0.8t+98.8\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dt}\\left(0.1t^{2}\\right)+\\frac{d}{dt}\\left(0.8t\\right)+\\frac{d}{dt}\\left(98.8\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(0.1t^{2}\\right)=0.2t$$", "input": "\\frac{d}{dt}\\left(0.1t^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=0.1\\frac{d}{dt}\\left(t^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=0.1\\cdot\\:2t^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=0.2t", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYn/JX5R/bR0Jv+MB4QOKmWX8zeERICEnv1Ds5A1/BdIwtOm6hKQUv9KoRlVVCqIipS1jTmJ+4y2m0/MCnsa/vPZrZTWPvyYGQkIcGcqdr+pVOCtbpgq2hgT+P329pNFEBCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(0.8t\\right)=0.8$$", "input": "\\frac{d}{dt}\\left(0.8t\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=0.8\\frac{dt}{dt}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dt}{dt}=1$$", "result": "=0.8\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=0.8", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmf6uv46fyI5d8YtbnjrDg6cWM5XpXh7YOsEqpbUSEW7MXlrEtOfJtbaSHXGd/GNrkeCBKuYKgaNJ253gLI69U6vFdXAKne0yK+D6TYOP4WhPNq0BBt0RzvbvcGzUmda+w==" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(98.8\\right)=0$$", "input": "\\frac{d}{dt}\\left(98.8\\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+idP5vLVrjUll6eMdYmYy6NptK28zfDyjv7VSK9KcWM5XpXh7YOsEqpbUSEW7ZuJKdCFsPJy1+5gBMEc9dg4bfwiV6iMLJ5sC1nL7dOY8R7kGsAxqJBPJ2i2OxTzssIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=-0.2t+0.8+0" }, { "type": "step", "primary": "Simplify", "result": "=-0.2t+0.8", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Increasing$$:-\\infty\\:<t<4,\\:$$Decreasing$$:4<t<\\infty\\:$$", "input": "f\\:{^{\\prime}}\\left(t\\right)=-0.2t+0.8", "steps": [ { "type": "interim", "title": "Find the critical points:$${\\quad}t=4$$", "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": "$$f\\:{^{\\prime}}\\left(t\\right)=0:{\\quad}t=4$$", "input": "-0.2t+0.8=0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$10$$", "input": "-0.2t+0.8=0", "result": "-2t+8=0", "steps": [ { "type": "step", "primary": "To eliminate decimal points, multiply by 10 for every digit after the decimal point", "secondary": [ "There is one digit to the right of the decimal point, therefore multiply by $$10$$" ], "result": "-0.2t\\cdot\\:10+0.8\\cdot\\:10=0\\cdot\\:10" }, { "type": "step", "primary": "Refine", "result": "-2t+8=0" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"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" } }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2t=-8", "result": "t=4", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2t}{-2}=\\frac{-8}{-2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-2t}{-2}=\\frac{-8}{-2}", "result": "t=4", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-2t}{-2}:{\\quad}t$$", "input": "\\frac{-2t}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2t}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=t" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dy1zB2bvBMuqVjwhgvaJZHyRHuGw7+tM5METTDj6vVHgWIADOS+aC1pY3tecNFb1ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz19zLufld8iOAWupCDwWGsEialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{-8}{-2}:{\\quad}4$$", "input": "\\frac{-8}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{8}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{2}=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bPLOPcK0KHndixDu4WUE5ACWKUbvV6WK3fDUgFtg3Q92ifU6Gu+Jf7ON5jwsOLEfo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCA72u0anm+xALIOHUJHBiptvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "result": "t=4" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "t=4" } ], "meta": { "interimType": "Explore Function Slope Zero Title 0Eq" } }, { "type": "interim", "title": "$$f\\:{^{\\prime}}\\left(t\\right)>0:{\\quad}t<4$$", "input": "-0.2t+0.8>0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$10$$", "input": "-0.2t+0.8>0", "result": "-2t+8>0", "steps": [ { "type": "step", "primary": "To eliminate decimal points, multiply by 10 for every digit after the decimal point", "secondary": [ "There is one digit to the right of the decimal point, therefore multiply by $$10$$" ], "result": "-0.2t\\cdot\\:10+0.8\\cdot\\:10>0\\cdot\\:10" }, { "type": "step", "primary": "Refine", "result": "-2t+8>0" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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"secondary": [ "There is one digit to the right of the decimal point, therefore multiply by $$10$$" ], "result": "-0.2t\\cdot\\:10+0.8\\cdot\\:10<0\\cdot\\:10" }, { "type": "step", "primary": "Refine", "result": "-2t+8<0" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$8\\:$$to the right side", "input": "-2t+8<0", "result": "-2t<-8", "steps": [ { "type": "step", "primary": "Subtract $$8$$ from both sides", "result": "-2t+8-8<0-8" }, { "type": "step", "primary": "Simplify", "result": "-2t<-8" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Multiply both sides by $$-1$$", "input": "-2t<-8", "result": "2t>8", "steps": [ { "type": "step", "primary": "Multiply both sides by -1 (reverse the inequality)", "result": "\\left(-2t\\right)\\left(-1\\right)>\\left(-8\\right)\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "2t>8" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2t>8", "result": "t>4", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2t}{2}>\\frac{8}{2}" }, { "type": "step", "primary": "Simplify", "result": "t>4" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "Combine intervals with domain", "result": "-\\infty\\:<t<4,\\:t=4,\\:4<t<\\infty\\:", "steps": [ { "type": "interim", "title": "Domain of 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