Find the Maximum and Minimum of $f(x)=x^3-x^2-8x+1 $ on $[-2,2]$
I take the first derivative which is: $f'(x)=3x^2-2x-8$
Since I am unable to determine the critical points from the first derivative, do I take the second derivative to find the critical points?
just because you are doing calculus doesn't mean you can forget all your precalculus! $$ 3x^2-2x-8=0\implies (3x+4)(x-2)=0 $$ Which will yield the critical points. Again, don't forget to check end points!
edit: from comments, I realize the above may be unclear. For a function you can differentiate on a interval which includes its endpoints, maxima and minima occur when the derivative is zero (think about what a horizontal tangent line looks like) or at end points.