Minimum and Maximum Values — Question 3

Question 3

Problem:

Consider the function $f(x) = x^4 - 4x^3 + 10$ on the interval $[0, 4]$.

• Find the critical points of $f(x)$ on the interval.

• Determine the absolute maximum and minimum values of $f(x)$ on the interval $[0, 4]$.

Question 3 - Solution

(a) Find critical points:

First, compute the derivative: $$f'(x) = 4x^3 - 12x^2$$

Factor: $$f'(x) = 4x^2(x - 3) \Rightarrow f'(x) = 0 \text{ when } x = 0 \text{ or } x = 3$$

Both are within the interval $[0, 4]$, so check both.

(b) Evaluate function at critical points and endpoints:

$$f(0) = 0 - 0 + 10 = 10$$ $$f(3) = 81 - 108 + 10 = -17$$ $$f(4) = 256 - 256 + 10 = 10$$

Conclusion:

- Minimum value: $\boxed{-17}$ at $x = 3$ - Maximum value: $\boxed{10}$ at $x = 0$ and $x = 4$

Graph of the function on $[0, 4]$: