Question

The critical points of the function \( f(x) = (x-3)^3(x+2)^2 \) are:

• A. \(-1, 3, -2\)
• B. \(1, 3, -2\)
• C. \(3, 3, -2\)
• D. \(0, 3, -2\)
• E. \(0, -3, 2\)

Hint:

When finding critical points, ensure to factorize the derivative completely to identify all points where the derivative is zero or the function is undefined.

Answer 1

The Correct Option is D

To find the critical points of \( f(x) \), we need to determine where the derivative \( f'(x) \) is equal to zero or undefined. First, calculate the derivative using the product rule: \[ f(x) = (x-3)^3(x+2)^2 \] \[ f'(x) = 3(x-3)^2(x+2)^2 + 2(x-3)^3(x+2) \] Simplify the derivative: \[ f'(x) = (x-3)^2(x+2)[3(x+2) + 2(x-3)] \] \[ = (x-3)^2(x+2)(3x + 6 + 2x - 6) \] \[ = (x-3)^2(x+2)(5x) \] \[ = 5x(x-3)^2(x+2) \] Set \( f'(x) \) equal to zero: \[ 5x(x-3)^2(x+2) = 0 \] This gives us three solutions: \[ x = 0, \quad x = 3, \quad x = -2 \] These are the points where the derivative is zero, indicating potential critical points.