Question

Use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) on the same set of coordinate axes and state the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) \(f(x)=x^{2} \sqrt{6-x^{2}},[-\sqrt{6}, \sqrt{6}]\)

Short answer

The first derivative of \(f(x) = 2x\sqrt{6-x^{2}} - \frac{x^{3}}{\sqrt{6-x^{2}}}\) and second derivative is \(f''(x)=-x^{2}\left(\frac{3x^{2}-12}{(6-x^{2})^{3/2}}\right). The relative extrema and points of inflection will be found by solving for x when these derivatives equate to zero. The graphs will depict these.

Step-by-step solution

Find the first derivative

To find the first derivative, we use the product rule and chain rule for differentiation. The product rule states that \((uv)' = u'v + uv'\) and chain rule states that \(f(g(x))'=f'(g(x))g'(x)\).\n The first derivative \(f'(x)\) of the function \(f(x)=x^{2} \sqrt{6-x^{2}}\) is obtained as follows: \(f'(x) = 2x\sqrt{6-x^{2}} + x^{2}\left(-\frac{x}{\sqrt{6-x^{2}}}\right) = 2x\sqrt{6-x^{2}} - \frac{(x^{3})}{\sqrt{6-x^{2}}}\)

Find the second derivative

To find the second derivative, we require to differentiate the first derivative obtained in Step 1 again. The obtained second derivative \(f''(x)\) will be: \(f''(x)=-\frac{3x^{2}}{(6-x^{2})^{3/2}}+ \frac{2}{(6-x^{2})^{1/2}} - 2x^{2}\left(\frac{-x}{(6-x^{2})^{3/2}}\right)= -x^{2}\left(\frac{3x^{2}-12}{(6-x^{2})^{3/2}}\)\)

Find relative extrema and points of inflection

To find the relative extrema and points of inflection, we need to solve \(f'(x) = 0\) and \(f''(x) = 0\) respectively, within the given interval [-\(\sqrt{6}\), \(\sqrt{6}\)]. The roots of these equations will be the x-coordinates of the extrema and inflection points.

Graph \(f(x)\), \(f'(x)\), and \(f''(x)\)

Plot the function \(f(x)\), the first derivative \(f'(x)\) and the second derivative \(f''(x)\) on the same set of coordinate axes. The function is increasing where \(f'(x) > 0\) and decreasing where \(f'(x) < 0\). The function has a local maximum at x where \(f'(x)= 0\) changes from positive to negative, and has a local minimum where \(f'(x)= 0\) changes from negative to positive. The function has an inflection point at x where \(f''(x) = 0\) changes sign.