Absolute extrema on a closed interval

Problem 3.245 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(x^{3} + 6 x^{2} + 9 x\right) = \left(x + 3\right) \left(3 x + 3\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-3, 5]: -1.
   Reviewed
4. \[ \left. x^{3} + 6 x^{2} + 9 x \right|_{\substack{ x=-3 }} = 0 \]
   f(-3).✓ Proved
5. \[ \left. x^{3} + 6 x^{2} + 9 x \right|_{\substack{ x=-1 }} = -4 \]
   f(-1).✓ Proved
6. \[ \left. x^{3} + 6 x^{2} + 9 x \right|_{\substack{ x=5 }} = 320 \]
   f(5).✓ Proved
7. The largest value is 320 and the smallest is -4.
   Reviewed