Question

For the function \(f(x) = x^3 - 3x\) on \([-1, 3]\), the points of maxima and minima are respectively:

• A. -1, 3
• B. 1, 3
• C. 1, -1
• D. -1, 1

Hint:

Exam Tip:
For local extrema:

• Use the first derivative test.
• Check where \(f'\) changes sign.
• For absolute extrema on a closed interval, also check endpoints.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to find the points of maxima and minima of a function on a closed interval. We must consider both critical points and endpoints.

Step 2: Key Formula or Approach:
Find the derivative \(f'(x)\), set it to zero to find critical points. Then evaluate the function at critical points and endpoints.

Step 3: Detailed Explanation:
\(f(x) = x^3 - 3x\). \[ f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x - 1)(x + 1) \] Set \(f'(x) = 0\):
\(3(x - 1)(x + 1) = 0 \Rightarrow x = 1\) or \(x = -1\).
Both critical points are in the interval \([-1, 3]\).
Evaluate \(f(x)\) at these points and the endpoints:
• \(f(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2\)
• \(f(1) = 1^3 - 3(1) = 1 - 3 = -2\)
• \(f(3) = 27 - 9 = 18\) The maximum value on the interval is 18 at \(x = 3\) (endpoint).
The minimum value is -2 at \(x = 1\) (critical point).
But the question asks for points of maxima and minima respectively.
The options are:
(A) -1, 3
(B) 1, 3
(C) 1, -1
(D) -1, 1
The maximum occurs at \(x = 3\) and the minimum occurs at \(x = 1\).
So, maxima = 3, minima = 1.
But the options have maxima first, then minima.
Option (A) gives maxima = -1, minima = 3 (not correct).
Option (B) gives maxima = 1, minima = 3 (not correct).
Option (C) gives maxima = 1, minima = -1 (not correct).
Option (D) gives maxima = -1, minima = 1 (not correct).
None of the options match.
Let's reconsider: The question might be asking for the points of local maxima and local minima (not absolute).
Local maxima occurs at \(x = -1\) (since \(f'\) changes from positive to negative).
Local minima occurs at \(x = 1\) (since \(f'\) changes from negative to positive).
So, maxima = -1, minima = 1.
This matches option (D).

Step 4: Final Answer:
Therefore, option (D) is correct.