Question

If the function $f(x) = x^3 - 3x^2 + 6$ is defined on the interval $[0,4]$, the maximum value is

• A. 6
• B. 2
• C. 22
• D. 18

Hint:

To find extreme values on a closed interval, always check endpoints and critical points.

Answer 1

The Correct Option is C

Step 1: Differentiate the function.
\[ f'(x) = 3x^2 - 6x \]
Step 2: Find critical points.
\[ 3x(x - 2) = 0 \Rightarrow x = 0, 2 \]
Step 3: Evaluate $f(x)$ at critical points and endpoints.
\[ f(0) = 6,\quad f(2) = 2,\quad f(4) = 22 \]
Step 4: Identify maximum value.
The maximum value on $[0,4]$ is $22$.