Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=\left(x^{2}+10\right)\left(100-x^{2}\right) \)

**Domain and Symmetry** - The function \[ f(x) = \left(x^2+10\right)\left(100-x^2\right) = -x^4+90x^2+1000 \] is defined for all real numbers, i.e. its domain is \( \mathbb{R} \). - Since the formula depends only on \( x^2 \), the function is even: \( f(-x)=f(x) \). This implies symmetry about the \( y \)-axis. **Intercepts** - \( y \)-intercept: \[ f(0) = (0^2+10)(100-0^2)=10\times 100 = 1000. \] So the point is \((0,1000)\). - \( x \)-intercepts: Set \[ (x^2+10)(100-x^2)=0. \] The factor \( x^2+10=0 \) gives no real solution. The equation \( 100-x^2=0 \) yields \[ x^2=100\quad\Longrightarrow\quad x=\pm 10. \] Thus the \( x \)-intercepts are \( (-10,0) \) and \( (10,0) \). **End Behavior** - The highest degree term is \( -x^4 \). Therefore, as \[ x\to\pm\infty,\quad -x^4\to-\infty, \] so \[ f(x)\to -\infty. \] **Critical Points and Local Extrema** 1. **Finding the derivative:** \[ f(x)=-x^4+90x^2+1000. \] The first derivative is \[ f'(x)=-4x^3+180x=-4x(x^2-45). \] 2. **Critical points:** \[ f'(x)=0\quad \Longrightarrow\quad x=0 \quad\text{or}\quad x^2=45,\quad x=\pm 3\sqrt{5}. \] 3. **Function values at the critical points:** - At \( x=0 \): \[ f(0)=1000. \] - At \( x=3\sqrt{5} \) (and by symmetry at \( x=-3\sqrt{5} \)): First note that \[ (3\sqrt{5})^2=45 \quad \text{and} \quad (3\sqrt{5})^4=(45)^2=2025. \] Then, \[ f(3\sqrt{5})=-2025+90\cdot 45+1000=-2025+4050+1000=3025. \] 4. **Conclusion:** - Local minima: At \((0,1000)\). - Local maxima: At \((3\sqrt{5}, 3025)\) and \(( -3\sqrt{5}, 3025)\). **Inflection Points** 1. **Second derivative:** \[ f''(x)=-12x^2+180=-12\bigl(x^2-15\bigr). \] 2. **Setting \( f''(x)=0 \):** \[ x^2-15=0\quad \Longrightarrow\quad x=\pm\sqrt{15}. \] 3. **Thus**, there are inflection points at \( x=\pm \sqrt{15} \). For example, at \( x=\sqrt{15} \): \[ f(\sqrt{15})=\bigl(15+10\bigr)\bigl(100-15\bigr)=25\cdot 85=2125. \] **Graph Characteristics Summary** - **Domain:** \( \mathbb{R} \) - **Symmetry:** Even (symmetric about the \( y \)-axis). - **\( y \)-intercept:** \((0, 1000)\). - **\( x \)-intercepts:** \((-10,0)\) and \((10,0)\). - **End Behavior:** As \( x\to \pm \infty \), \( f(x)\to -\infty \). - **Local Extremum:** - Local minimum at \((0, 1000)\). - Local maxima at \((3\sqrt{5}, 3025)\) and \(( -3\sqrt{5}, 3025)\). - **Inflection Points:** At \( x=\pm \sqrt{15} \) (e.g. \( (\sqrt{15},2125) \) and by symmetry at \( (-\sqrt{15},2125) \)). **Sketch Guidance** - Plot the intercepts: mark \((0,1000)\) on the \( y \)-axis and \((-10, 0)\), \((10, 0)\) on the \( x \)-axis. - Plot the local maxima at approximately \( x \approx \pm 6.708 \) (since \( 3\sqrt{5}\approx 6.708 \)) with \( y=3025 \). - Note the local minimum at the origin’s vertical line \((0, 1000)\). - Mark the inflection points at \( x\approx \pm 3.873 \) (since \( \sqrt{15}\approx 3.873 \)) with \( y\approx2125 \). - Since the function is even, reflect the graph about the \( y \)-axis. - Finally, draw the tails dropping to \(-\infty\) as \( |x| \) increases beyond the intercepts at \(\pm10\). This completes the summary of the pertinent information and provides the main features needed to sketch the graph of \[ f(x)=\left(x^2+10\right)\left(100-x^2\right). \]