Question

The absolute maximum of the function \(f(x) = x^{3 - 3x + 2\) in the interval \([0, 2]\) is ________}

• A. 0
• B. 2
• C. 4
• D. 6

Hint:

Always check whether critical points lie inside the given interval.
In this problem, $x = -1$ is a critical point but is excluded because it lies outside $[0, 2]$.
Evaluating only the relevant points saves time and prevents errors.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find the absolute maximum value of a cubic polynomial function over a closed and bounded interval \([0, 2]\).

Step 2: Key Formula or Approach:
According to the Extreme Value Theorem, the absolute extrema of a continuous function on a closed interval occur either at its critical points within the interval or at the endpoints of the interval.
1. Find the first derivative \(f'(x)\) and solve \(f'(x) = 0\) to locate critical points.
2. Evaluate \(f(x)\) at these critical points and at the endpoints \(x = 0\) and \(x = 2\).
3. Identify the largest value from these results.

Step 3: Detailed Explanation:

•
Step 3.1: Locate the critical points:
Find the derivative of \(f(x)\):
\[ f'(x) = 3x^{2} - 3 \] Set the derivative to zero:
\[ 3x^{2} - 3 = 0 \implies x^{2} = 1 \implies x = \pm 1 \] Only \(x = 1\) lies within the given interval \([0, 2]\).

•
Step 3.2: Evaluate the function at critical points and endpoints:
At the left endpoint \(x = 0\):
\[ f(0) = (0)^{3} - 3(0) + 2 = 2 \] At the critical point \(x = 1\):
\[ f(1) = (1)^{3} - 3(1) + 2 = 1 - 3 + 2 = 0 \] At the right endpoint \(x = 2\):
\[ f(2) = (2)^{3} - 3(2) + 2 = 8 - 6 + 2 = 4 \]
•
Step 3.3: Compare values:
The values of the function are: \(f(0) = 2\), \(f(1) = 0\), and \(f(2) = 4\).
The largest of these values is 4.

Step 4: Final Answer:
The absolute maximum of the function in the interval \([0, 2]\) is 4.