Absolute extrema on a closed interval

Problem 3.199 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(x^{3} + \frac{9 x^{2}}{2} + 6 x + 1\right) = \left(x + 2\right) \left(3 x + 3\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-2, 1]: -1.
   Reviewed
4. \[ \left. x^{3} + \frac{9 x^{2}}{2} + 6 x + 1 \right|_{\substack{ x=-2 }} = -1 \]
   f(-2).✓ Proved
5. \[ \left. x^{3} + \frac{9 x^{2}}{2} + 6 x + 1 \right|_{\substack{ x=-1 }} = - \frac{3}{2} \]
   f(-1).✓ Proved
6. \[ \left. x^{3} + \frac{9 x^{2}}{2} + 6 x + 1 \right|_{\substack{ x=1 }} = \frac{25}{2} \]
   f(1).✓ Proved
7. The largest value is 25/2 and the smallest is -3/2.
   Reviewed