Absolute extrema on a closed interval

Problem 3.359 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(- x^{3} + \frac{3 x^{2}}{2} - 4\right) = - 3 x \left(x - 1\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-2, 3]: 0, 1.
   Reviewed
4. \[ \left. - x^{3} + \frac{3 x^{2}}{2} - 4 \right|_{\substack{ x=-2 }} = 10 \]
   f(-2).✓ Proved
5. \[ \left. - x^{3} + \frac{3 x^{2}}{2} - 4 \right|_{\substack{ x=0 }} = -4 \]
   f(0).✓ Proved
6. \[ \left. - x^{3} + \frac{3 x^{2}}{2} - 4 \right|_{\substack{ x=1 }} = - \frac{7}{2} \]
   f(1).✓ Proved
7. \[ \left. - x^{3} + \frac{3 x^{2}}{2} - 4 \right|_{\substack{ x=3 }} = - \frac{35}{2} \]
   f(3).✓ Proved
8. The largest value is 10 and the smallest is -35/2.
   Reviewed