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Math Help - exact local max

  1. #1
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    exact local max

    Let f(x)=x^80/e^x= x^80*e^-x. A graph of y= f(x) is shown below. There is clearly a local maximum point somewhere around x= 75 (actually the extree point is to the right of x=75). Also there appears to be at least one local minimum between x=-15 and x=45.

    Use calculus to find the exact location of the local maximum that is near x=75 show that there is a local minimum at x=0.

    I cannot show the grow however i can tell you that the y range goes by .5*10^116. However after that the exponents remain at 117 rather than 116. The x range goes by 15 starting at -30 going up to 150. Also I can see that the local max according to the graph is at coordinates (80, 3.1*10^117).

    How would i figure this out? Would I find the slope of the tangent line?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by asweet1 View Post
    Let f(x)=x^80/e^x= x^80*e^-x. A graph of y= f(x) is shown below. There is clearly a local maximum point somewhere around x= 75 (actually the extree point is to the right of x=75). Also there appears to be at least one local minimum between x=-15 and x=45.

    Use calculus to find the exact location of the local maximum that is near x=75 show that there is a local minimum at x=0.

    I cannot show the grow however i can tell you that the y range goes by .5*10^116. However after that the exponents remain at 117 rather than 116. The x range goes by 15 starting at -30 going up to 150. Also I can see that the local max according to the graph is at coordinates (80, 3.1*10^117).

    How would i figure this out? Would I find the slope of the tangent line?
    f(x)=x^{80}e^{-x}
    f'(x)=80x^{79}e^{-x}-x^{80}e^{-x}=x^{79}e^{-x}(80-x)

    Set f'(x)=0 and solve for x. You can use the second derivative test to see if it's a max or min:

    f''(x_0)>0\implies\min
    f''(x_0)<0\implies\max
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  3. #3
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    so would i plug the derivative into my calculator and try to find the local max
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by asweet1 View Post
    so would i plug the derivative into my calculator and try to find the local max
    You don't need a calculator to find the extrema. A calculator might be helpful to see whether they're local mins or maxes, but that's it.

    x^{79}e^{-x}(80-x)=0\implies x^{79}=0~\text{or}~80-x=0\implies x=0~\text{or}~x=80
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