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

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UpStudy AI · Accepted Answer

**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. $$ y=1000 $$.
**Summary of the Function $$ f(x) = (x^{2} + 10)(100 - x^{2}) $$:** - **Domain:** All real numbers, $$ (-\infty, \infty) $$. - **$$ x $$-Intercepts:** $$ x = -10 $$ and $$ x = 10 $$. - **$$ y $$-Intercept:** $$ y = 1000 $$. **Graph Characteristics:** - The function is even and symmetric about the $$ y $$-axis. - As $$ |x| $$ increases, $$ f(x) $$ approaches negative infinity. - Local maximum at $$ x = \pm 3\sqrt{5} $$ with $$ f(x) = 3025 $$. - Local minimums at the ends of the domain with $$ f(x) \to -\infty $$. **Answer Selections:** - **Domain:** A. $$ (-\infty, \infty) $$. - **$$ x $$-Intercepts:** A. $$ x = -10, \, 10 $$. - **$$ y $$-Intercept:** A. $$ y = 1000 $$.

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