Find the intervals on which \( f(k) \) is increasing anls the inter \( I(x)=x^{4}-50 x^{2} \) Select the correct choice below and, if necessary, fil in th A. The function is incleasing on \( \square \). (Type your answer using nterval nolation. Use a B. The function is never increasing Select the correct choice below and, if necessary, fill in in A. The funclion is decreasing on \( \square \). (Type your answer using intorval nolation, Use a B. The funclion is never decieasing.

To determine where the function \( I(x) = x^4 - 50x^2 \) is increasing or decreasing, we first need to find its derivative: \[ I'(x) = 4x^3 - 100x \] Setting the derivative equal to zero to find critical points: \[ 4x(x^2 - 25) = 0 \implies x = 0, x = 5, x = -5 \] Next, we test the intervals given by these critical points: \( (-\infty, -5) \), \( (-5, 0) \), \( (0, 5) \), and \( (5, \infty) \). 1. For \( x < -5 \), say \( x = -6 \): \[ I'(-6) = 4(-6)^3 - 100(-6) > 0 \rightarrow \text{Increasing} \] 2. For \( -5 < x < 0 \), say \( x = -1 \): \[ I'(-1) = 4(-1)^3 - 100(-1) < 0 \rightarrow \text{Decreasing} \] 3. For \( 0 < x < 5 \), say \( x = 1 \): \[ I'(1) = 4(1)^3 - 100(1) < 0 \rightarrow \text{Decreasing} \] 4. For \( x > 5 \), say \( x = 6 \): \[ I'(6) = 4(6)^3 - 100(6) > 0 \rightarrow \text{Increasing} \] Thus, the function is increasing on the intervals \( (-\infty, -5) \) and \( (5, \infty) \). Additionally, the function is decreasing on the interval \( (-5, 5) \). So, for your choices: A. The function is increasing on \( (-\infty, -5) \) and \( (5, \infty) \). A. The function is decreasing on \( (-5, 5) \).