Answer to Question #17672 in Calculus for hsd

Question

Find the absolute maximum and minimum values of the following function on the given interval. If there are multiple points in a single category list the points in increasing order in x value and enter N in any blank that you don't need to use.

f(x)=8e^(−x)−8e^(−2x) , [0,1]

Absolute maxima
x = y =
x = y =
x = y =

Absolute minima
x = y =
x = y =
x = y =

Answer 1

we can check only ends of interval and points where f'=0
f'(x)=-8e^(-x)+16e^(-2x)
f'=0
16e^(-2x)=8e^(-x)
e^(-x)>0 so
2e^(-x)=1
e^(-x)=1/2
x=-ln(1/2)=ln(2)

x=0 f(x)=8-8=0
x=ln(2) f(x)=8e^(-ln(2))-8e^(-ln(4))=4-2=2
x=1 f(x)=8(1/e-1/e^2)
absolute minima x=0 y=0
absolute maxima x=ln(2) y=2