The function \( f(x)=-x^{4}+72 x^{2} \) has A. one relative minimum and two relative maxima. B. one relative minimum and one relative maximum. C. no relative minimum and two relative maxima. D. two relative minima and no relative maximum. E. two relative minima and one relative maximum.

To analyze the function \( f(x) = -x^4 + 72x^2 \), we first find the critical points by calculating its derivative: \[ f'(x) = -4x^3 + 144x \] Setting the derivative to zero gives us: \[ -4x^3 + 144x = 0 \] \[ 4x(-x^2 + 36) = 0 \] This results in: \[ x = 0 \quad \text{or} \quad x^2 = 36 \quad \Rightarrow \quad x = \pm 6 \] Next, we determine the nature of these critical points using the second derivative: \[ f''(x) = -12x^2 + 144 \] Evaluating the second derivative at the critical points: - For \( x = 0 \): \[ f''(0) = 144 > 0 \] (relative minimum) - For \( x = 6 \): \[ f''(6) = -12(6^2) + 144 = -432 + 144 = -288 < 0 \] (relative maximum) - For \( x = -6 \): \[ f''(-6) = -12(-6^2) + 144 = -432 + 144 = -288 < 0 \] (relative maximum) Therefore, the function has **one relative minimum at \( x = 0 \)** and **two relative maxima at \( x = 6 \) and \( x = -6 \)**, which corresponds to option A: one relative minimum and two relative maxima.