Absolute extrema on a closed interval

Problem 3.205 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
2. \[ \frac{d}{d x} \left(x^{3} - 6 x^{2} + 9 x + 3\right) = \left(x - 1\right) \left(3 x - 9\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-1, 3]: 1.
4. \[ \left. x^{3} - 6 x^{2} + 9 x + 3 \right|_{\substack{ x=-1 }} = -13 \]
   f(-1).✓ Proved
5. \[ \left. x^{3} - 6 x^{2} + 9 x + 3 \right|_{\substack{ x=1 }} = 7 \]
   f(1).✓ Proved
6. \[ \left. x^{3} - 6 x^{2} + 9 x + 3 \right|_{\substack{ x=3 }} = 3 \]
   f(3).✓ Proved
7. The largest value is 7 and the smallest is -13.