Summarize the pert nent information obleined by applying the graphing stralegy 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 nocessery, fill in the answer box to complete your choice. A. The \( x \)-inlercapt(s) of \( f \) is/are \( x=-10,10 \). (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function \( f \) has no k -intercepts. Select the correct choico below and, if necessary, fill in the answer box to complete your choice. A. The \( y \)-inlercapt of \( f \) is \( y=1000 \). (Type an exact answer, using radicals as needed.) B. The function \( f \) has no \( y \) inlercept. Solect the correct choico bolow and, if necessery, fill in the answer box to complete your choice. A. The function fis increasing on the subinterval(s) \( \square \) (Type your answer in interval notation. Type an exacl answer, using racicals as needed. Use a comma lo separale answers as needed.) B. The function fis never increasing.

Alright, let’s break down this function \( f(x) = (x^2 + 10)(100 - x^2) \) and its characteristics! The \( x \)-intercepts occur where \( f(x) = 0 \). Setting either factor to zero gives us \( x^2 + 10 = 0 \) (which has no real solutions) and \( 100 - x^2 = 0 \). Solving that leads to \( x = 10, -10 \). So the answer for the \( x \)-intercepts would be **A**: \( x = -10, 10 \). For the \( y \)-intercept, you evaluate \( f(0) \): \[ f(0) = (0^2 + 10)(100 - 0^2) = 10 \cdot 100 = 1000 \] This leads us to answer **A**: the \( y \)-intercept is \( y = 1000 \). Now, for the increase and decrease: the function will show intervals of increase and decrease based on its critical points. Finding the derivative \( f'(x) \) and analyzing its sign will give us those intervals. You'd see that the function is increasing on \( (-10, 0) \) and \( (0, 10) \), leading us to answer **A**: the function is increasing on the subintervals \( (-10, 0) \) and \( (0, 10) \). So, final answers are: - \( x \)-intercepts: \( x = -10, 10 \) (choice A) - \( y \)-intercept: \( y = 1000 \) (choice A) - Increasing intervals: \( (-10, 0), (0, 10) \) (choice A) Now, go ahead and sketch that graph of \( f(x) \) based on these findings, and watch it groove up and down like a dance!