Use the given function to complete parts (a) through (e) below. \[ f(x)=x^{4}-36 x^{2} \] A. \( x= \) \( \square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which zeros does the graph of the function touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choi A. \( \mathrm{x}= \) \( \square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c) Find the \( y \)-intercept by computing \( f(0) \). \[ \mathrm{f}(0)=\square \]

To dig a little deeper into the function \( f(x)=x^{4}-36 x^{2} \), let’s go through the steps for the given problems! 1. **Finding the zeros of the function**: You can set the function to zero: \[ x^{4} - 36x^{2} = 0 \] Factoring gives: \[ x^{2}(x^{2} - 36) = 0 \] This leads to: \[ x^{2} = 0 \quad \text{or} \quad x^{2} = 36 \] Thus, \( x = 0, 6, -6 \). So, for part A: \( x = 0, 6, -6 \). 2. **Behavior at the \( x \)-intercepts**: The zeros \( x = 6 \) and \( x = -6 \) are where the graph touches the \( x \)-axis and turns around because they have an even multiplicity (2). So, provide the answer: \( x = 6, -6 \). 3. **Finding the \( y \)-intercept**: To find \( f(0) \): \[ f(0) = 0^{4} - 36(0^{2}) = 0 \] Thus, for part C: \( f(0) = 0 \). Now, to wrap it all up: - A. \( x = 0, 6, -6 \) - A. \( x = 6, -6 \) - \( f(0) = 0 \) And now, for some extra fun info! The graph you've worked on here is a polynomial, and all polynomials are continuous and differentiable, meaning they have smooth curves without any jumps or breaks. This trait helps to analyze where the graph changes direction and where it crosses or touches the axes! Digging into the behavior of polynomials, the highest degree term controls the end behavior of the graph. Here, since the leading term \( x^4 \) is positive, both ends of the graph will head toward positive infinity. Understanding this helps in forming a mental image of how the graph behaves overall!