Introduction to extrema

Use WolframAlpha to accomplish the following:

Let \( f(x) = \frac{1}{5} x^5 + x^4 -4 x^3 + 3 \) on \( [-8, 4] \).

1. Find the "critical numbers" of \( f(x) \). Answer
2. Plot \( f'(x) \). What happens to the derivative at the critical numbers? Answer
3. Find the extrema of \( f(x) \) with their locations. Answer
4. How do the locations of the extrema compare to the critical numbers?
5. Notice an unusual result when \( x=0 \). Answer
6. Notice an unusual result when \( x=-6 \). Answer