Finding Absolute Extrema — Question 1

Question 1 - Solution

We are given: $$f(x) = x^3 - 6x^2 + 9x + 2$$

Step 1: Find critical points

$$f'(x) = 3x^2 - 12x + 9 = 3(x - 1)(x - 3) = 0 \Rightarrow x = 1, \ 3$$

Step 2: Evaluate at endpoints and critical points in $[0, 4]$

$$\begin{aligned} f(0) &= 2 \\ f(1) &= 6 \\ f(3) &= 2 \\ f(4) &= 6 \end{aligned}$$

Absolute Maximum: $\boxed{6}$ at $x = 1$ and $x = 4$

Absolute Minimum: $\boxed{2}$ at $x = 0$ and $x = 3$

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