Question

The function \( (x - 2)^2 (x + 2)^2 \) has

• A. minima at +2 and maxima at -2
• B. minima at -2 and maxima at +2
• C. minima at -2 and +2
• D. maxima at -2 and +2

Hint:

When analyzing the function, use the first and second derivative tests to determine the critical points and identify whether they correspond to maxima or minima.

Answer 1

The Correct Option is C

To find the maxima or minima of the function \( f(x) = (x - 2)^2 (x + 2)^2 \), we first expand the expression: \[ f(x) = [(x - 2)(x + 2)]^2 = (x^2 - 4)^2 \] Now, take the derivative to find the critical points: \[ f'(x) = 2(x^2 - 4) \cdot 2x = 4x(x^2 - 4) \] Set \( f'(x) = 0 \) to find the critical points: \[ 4x(x^2 - 4) = 0 \] This gives the critical points at \( x = 0, 2, -2 \). - For \( x = 2 \) and \( x = -2 \), the second derivative test shows that these points are minima (since \( f''(x)>0 \) at these points). - At \( x = 0 \), the function has a maximum since the function changes concavity. Therefore, the function has minima at \( x = -2 \) and \( x = +2 \), and hence the correct answer is (C) minima at \( -2 \) and \( +2 \), and (A) minima at \( +2 \) and maxima at \( -2 \).