Can you help me how to find z in this problem?

Question

Consider the function on .

Absolute Maximum value............ at =.............

Absolute Minimum value............. at =............

I just did the derivative which is r'(z)= -16e^(-4z) +4e^(-z)=0, but I don't know how to find z.

Answer 1

You are almost there!

r'(z)= -16e^(-4z) +4e^(-z)=4e^(-4z)(e^(3z)-4)=0, thus z=(1/3)*ln4=0.462.

r''(z=0.462)>0, thus at z=0.462, the minimum is achieved, which is r(z=0.462) = -1.8899

The maximum is achieved at z=0 or +∞, both has r(z)=0