Question #17672

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 =

Expert's answer

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

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

## Comments

## Leave a comment