Find local maxima and local minima for the function f(x) = x³ - 3x.

• A. Local max. at x = 1, local min. at x = -1
• B. Local max. at x = -1, local min. at x = 1
• C. Local max. at x = -1, no local min.
• D. Local min. at x = 1, no local max.

To find the local maxima and minima of the function f(x) = x³ - 3x, we first need to calculate the derivative and set it to zero. The derivative is:

Step 1: Find the Derivative

The derivative of the function is:

f'(x) = 3x² - 3

Step 2: Set the Derivative to Zero

Next, we set the derivative equal to zero to find critical points:

3x² - 3 = 0

Solving this gives:

• x² = 1
• x = 1 and x = -1

Step 3: Determine Local Maxima and Minima

To classify these critical points, we can use the second derivative test:

f''(x) = 6x

Now, evaluate the second derivative at the critical points:

• At x = 1: f''(1) = 6(1) = 6 (positive, indicating a local minimum)
• At x = -1: f''(-1) = 6(-1) = -6 (negative, indicating a local maximum)

Final Results

Thus, we conclude that:

• Local maximum at x = -1
• Local minimum at x = 1

The correct answer is A. Local max. at x = 1, local min. at x = -1.