Absolute extrema on a closed interval

Problem 3.281 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(x^{3} - 3 x - 2\right) = \left(x + 1\right) \left(3 x - 3\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-4, 4]: -1, 1.
   Reviewed
4. \[ \left. x^{3} - 3 x - 2 \right|_{\substack{ x=-4 }} = -54 \]
   f(-4).✓ Proved
5. \[ \left. x^{3} - 3 x - 2 \right|_{\substack{ x=-1 }} = 0 \]
   f(-1).✓ Proved
6. \[ \left. x^{3} - 3 x - 2 \right|_{\substack{ x=1 }} = -4 \]
   f(1).✓ Proved
7. \[ \left. x^{3} - 3 x - 2 \right|_{\substack{ x=4 }} = 50 \]
   f(4).✓ Proved
8. The largest value is 50 and the smallest is -54.
   Reviewed