Question

The absolute maximum value of function \( f(x) = x^3 - 3x + 2 \) in \( [0, 2] \) is:

• A. 0
• B. 2
• C. 4
• D. 5

Hint:

To find the absolute maximum or minimum, check the function values at critical points and endpoints within the given interval.

Answer 1

The Correct Option is C

To determine the absolute maximum of \( f(x) \) on \( [0, 2] \), we first identify critical points by differentiating \( f(x) \).
The derivative is \( f'(x) = 3x^2 - 3 \).
Setting \( f'(x) = 0 \) yields the critical points: \[3x^2 - 3 = 0 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = 1.\] Next, we evaluate \( f(x) \) at the interval endpoints and the critical point \( x = 1 \): - At \( x = 0 \): \( f(0) = 0^3 - 3(0) + 2 = 2 \) - At \( x = 1 \): \( f(1) = 1^3 - 3(1) + 2 = 0 \) - At \( x = 2 \): \( f(2) = 2^3 - 3(2) + 2 = 4 \) The absolute maximum value is \( 4 \), occurring at \( x = 2 \).