Absolute extrema on a closed interval

Problem 3.240 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(x^{3} - \frac{9 x^{2}}{2} + 6 x + 1\right) = \left(x - 1\right) \left(3 x - 6\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [0, 5]: 1, 2.
   Reviewed
4. \[ \left. x^{3} - \frac{9 x^{2}}{2} + 6 x + 1 \right|_{\substack{ x=0 }} = 1 \]
   f(0).✓ Proved
5. \[ \left. x^{3} - \frac{9 x^{2}}{2} + 6 x + 1 \right|_{\substack{ x=1 }} = \frac{7}{2} \]
   f(1).✓ Proved
6. \[ \left. x^{3} - \frac{9 x^{2}}{2} + 6 x + 1 \right|_{\substack{ x=2 }} = 3 \]
   f(2).✓ Proved
7. \[ \left. x^{3} - \frac{9 x^{2}}{2} + 6 x + 1 \right|_{\substack{ x=5 }} = \frac{87}{2} \]
   f(5).✓ Proved
8. The largest value is 87/2 and the smallest is 1.
   Reviewed