Can extreme values occur at values of x for which f prime of x doesn't equal 0?

Answer 1

Yes- for example, if the derivative is undefined there because there is a sharp corner, as in an absolute value function

Answer 2

Absolutely.

Take the function f(x) = sqrt(x).

f'(x) = 1/[2sqrt(x)]

Note that this has no solutions if f'(x) = 0. However, there is a minimum at x = 0 (at the point (0,0)), and you'd understand that if you're familiar with the graph itself.

That's why, when solving for the first derivative, you have to determine the critical points where f'(x) = 0 -OR- where f'(x) is undefined.

As a second example, take the function f(x) = |x|. As you may already know, this looks like a "V", with the bottom of the V at (0,0). If we take the derivative,

f'(x) = x/|x|

And, as you can see, this function is undefined at x = 0 but there is a minimum at x = 0.