1.      Find the absolute maximum and the absolute minimum of the function f(x) = xe-x on [-10, 10].  Show all working!

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aruv | High School Teacher | (Level 2) Valedictorian

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Find the absolute maximum and the absolute minimum of the function `f(x)=xe^(-x)` on [-10, 10]. 

For maxima and minima, first differentiate f(x) with respect to x.

`f'(x)=e^(-x)-xe^(-x)`

Solve

`e^(-x)(1-x)=0`

But `e^(-x)!=0AAx in [-10,10].`  Thus

1-x=0

x=1 and `x in[-10,10].`

`f''(x)=-e^(-x)-(e^(-x)-xe^(-x))`

`=-2e^(-x)+xe^(-x)`

`=e^(-x)(-2+x)`

`f''(1)=e^(-1)(-2+1)=-e^(-1)<0`

Thus f(x) has maximum value at x=1 which is absolute maxima.

Thus maximum f(x)=`e^(-1)` .

Since f(x) is continuous function [-10,10]. So it will be bounded function in [-10,10]. Absolute minima will be greatest lower bound of f(x) which will occure at x=-10. Thus absolute minimum value of  f(x) in [-10,10] will be `-10e^10.`  

 

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