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

**Domain and Symmetry** - The function is given by \[ f(x) = (x^2+10)(100-x^2). \] Since it is a polynomial, its domain is all real numbers. - Notice that \[ f(-x) = ((-x)^2+10)(100-(-x)^2) = (x^2+10)(100-x^2) = f(x), \] so \( f(x) \) is an even function (symmetric about the \( y \)-axis). --- **Intercepts** - **\( x \)-intercepts**: Set \( f(x)=0 \). Since \[ f(x) = (x^2+10)(100-x^2), \] the factor \( x^2+10 \) is always positive for real \( x \). Thus, the zeros come from \[ 100-x^2=0 \quad \Rightarrow \quad x^2=100 \quad \Rightarrow \quad x=\pm 10. \] - **\( y \)-intercept**: Compute \( f(0) \). \[ f(0) = (0^2+10)(100-0^2) = 10\times 100 = 1000. \] So the graph passes through \((0, 1000)\). --- **Rewriting the Function** Expanding gives \[ f(x) = (x^2+10)(100-x^2)= 100x^2 + 1000 - x^4 -10x^2 = -x^4 + 90x^2+1000. \] --- **Critical Points and Extrema** 1. **Finding the derivative:** \[ f(x) = -x^4+90x^2+1000 \quad \Rightarrow \quad f'(x) = -4x^3+180x. \] Factor the derivative: \[ f'(x) = -4x(x^2-45)= -4x(x-\sqrt{45})(x+\sqrt{45}). \] The critical points are at: \[ x=0 \quad \text{and} \quad x=\pm \sqrt{45} = \pm 3\sqrt{5}. \] 2. **Evaluating \( f(x) \) at the critical points:** - At \( x=0 \): \[ f(0)=1000. \] - At \( x=\pm 3\sqrt{5} \): First, compute \[ (3\sqrt{5})^2 = 9\times5=45 \quad \text{and} \quad (3\sqrt{5})^4 = 45^2=2025. \] Then, \[ f(3\sqrt{5}) = -2025+90(45)+1000 = -2025+4050+1000=3025. \] By symmetry, \( f(-3\sqrt{5}) = 3025 \). 3. **Interpreting the Critical Points:** - \( x=\pm 3\sqrt{5} \) are local maximum points with a maximum value of \( 3025 \). - \( x=0 \) is a local minimum within the interval between these local maxima, though note that the endpoints \( x=\pm 10 \) yield \( f(\pm10)=0 \) which are lower; however, those are not local extrema because the function is decreasing past \( |x|=10 \). --- **End Behavior** - The leading term in the expanded form \( f(x)=-x^4+90x^2+1000 \) is \(-x^4\). - Thus, as \( x\to\pm\infty \), \[ f(x) \to -\infty. \] --- **Graph Summary and Sketch Description** - **Symmetry:** The graph is even (symmetric about the \( y \)-axis). - **Key Points:** - \( x \)-intercepts: \( \; x=-10 \) and \( x=10 \) (where \( f(x)=0 \)). - \( y \)-intercept: \((0,1000)\). - Local maxima: \(\; x=\pm 3\sqrt{5} \approx \pm 6.708 \) with \( f(x)=3025 \). - Local minimum (within \(|x|<10\)): at \( (0, 1000) \). - For \(|x|>10\), the function becomes negative and falls toward \(-\infty\). - **Sketch Outline:** - Draw the \( y \)-axis as the axis of symmetry. - Mark the zeros at \((-10, 0)\) and \((10, 0)\). - Plot the \( y \)-intercept at \((0, 1000)\). - Plot the local maximum points at approximately \((-6.71, 3025)\) and \((6.71, 3025)\). - For \( |x| > 10 \), the curve drops below the \( x \)-axis tapering off to \(-\infty\). - Connect these points with a smooth curve, ensuring the symmetry and the proper end behavior as \( x\to\pm\infty \). --- **Conclusion** The graph of \[ f(x)=\left(x^{2}+10\right)\left(100-x^{2}\right) \] has the following pertinent features: - Domain: All real numbers. - Even symmetry about the \( y \)-axis. - \( x \)-intercepts at \( x=\pm 10 \). - \( y \)-intercept at \((0, 1000)\). - Local maximum points at \( x=\pm 3\sqrt{5} \) with \( f(x)=3025 \). - A local minimum at \( x=0 \) with \( f(x)=1000 \) (on the interval \([-10,10]\)). - As \( x\to\pm\infty \), \( f(x)\to -\infty \). This information guides the sketch of the graph, ensuring accuracy in the locations of intercepts, extrema, symmetry, and end behavior.