Summanze the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \[ f(x)=\left(x^{2}+10\right)\left(100-x^{2}\right) \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The domain of \( f \) is empty Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The x -intercept(s) of f is/are \( \mathrm{x}=-10,10 \) (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function \( f \) has no \( x \)-intercepts. Select the correct choice below and if necessary, fill in the answer box to complete your choice. A. The \( y \)-intercept of \( f \) is \( y= \) \( \square \) (Type an exact answer, using radicals as needed.) B. The function fhas no \( y \)-intercept

**Domain** The function is \[ f(x)=(x^{2}+10)(100-x^{2}). \] Since both \( x^2+10 \) and \( 100-x^2 \) are defined for all real numbers, the domain of \( f \) is \[ (-\infty, \infty). \] **X‑Intercepts** To find the \( x \)‑intercepts, set \( f(x)=0 \): \[ (x^2+10)(100-x^2)=0. \] This gives two possible equations: 1. \( x^2+10=0 \) \[ x^2=-10, \] which has no real solutions. 2. \( 100-x^2=0 \) \[ x^2=100 \quad \Rightarrow \quad x=\pm 10. \] Thus, the \( x \)‑intercepts occur at \( x=-10 \) and \( x=10 \). **Y‑Intercept** To find the \( y \)‑intercept, set \( x=0 \): \[ f(0)=(0^{2}+10)(100-0^{2})=10 \times 100=1000. \] So, the \( y \)‑intercept is \[ y=1000. \] **Graphing Strategy Summary** 1. **Domain:** \[ (-\infty, \infty) \] 2. **Intercepts:** - \( x \)‑intercepts: \( x=-10,\, 10 \) - \( y \)‑intercept: \( y=1000 \) 3. **Behavior and Symmetry:** - Expanding the function gives \[ f(x)=(x^2+10)(100-x^2)= -x^{4}+90x^{2}+1000. \] - The highest-degree term is \(-x^4\), so as \(|x| \to \infty\), \( f(x) \to -\infty \). - All powers of \( x \) in the expanded form are even; therefore, the function is even and symmetric about the \( y \)-axis. 4. **Critical Points (for detailed sketching):** - Differentiate: \[ f'(x)=-4x^3+180x. \] - Factor: \[ f'(x)=-4x(x^2-45)=0, \] yielding \( x=0 \) and \( x=\pm \sqrt{45}= \pm 3\sqrt{5} \). - Compute \( f(3\sqrt{5}) \): \[ (3\sqrt{5})^2=45, \quad (3\sqrt{5})^4=45^2=2025, \] \[ f(3\sqrt{5})=-2025+90\times 45+1000=-2025+4050+1000=3025. \] Thus, there is a local maximum at \( x=\pm 3\sqrt{5} \) with \( f(x)=3025 \) and a local minimum at the ends as \( f(x) \to -\infty \). **Answer Selections** - Domain: A. \( (-\infty, \infty) \). - \( x \)-intercepts: A. \( x=-10, \, 10 \). - \( y \)-intercept: A. \( y=1000 \).