91Ó°ÊÓ

The given function is \[f(x) = x^{\frac{5}{3}} + 5x^{\frac{2}{3}}\]. To find its derivative, we will apply the power rule of differentiation to each term: \[f'(x) = \frac{5}{3}x^{\frac{5}{3} - 1} + \frac{10}{3}x^{\frac{2}{3} - 1}\] \[f'(x) = \frac{5}{3}x^{\frac{2}{3}} + \frac{10}{3}x^{-\frac{1}{3}}\] Now we have the first derivative of the function.

Critical points occur when the derivative is either equal to zero or undefined. In this case, we need to solve the equation \(f'(x) = 0\): \[\frac{5}{3}x^{\frac{2}{3}} + \frac{10}{3}x^{-\frac{1}{3}} = 0\] To solve this equation, we can factor out \(\frac{5}{3}x^{-\frac{1}{3}}\): \[\frac{5}{3}x^{-\frac{1}{3}}(x + 2) = 0\] Now set each factor equal to zero: \[\frac{5}{3}x^{-\frac{1}{3}} = 0 \Rightarrow x = 0\] and \[x + 2 = 0 \Rightarrow x = -2\] Therefore, our critical points are at x = -2, 0.

To check for relative maxima or minima at the critical points, we can use the second derivative test. First, find the second derivative of the function: \[f''(x) = \frac{d^2f}{dx^2} = \frac{d}{dx}\left(\frac{5}{3}x^{\frac{2}{3}} + \frac{10}{3}x^{-\frac{1}{3}}\right) = -\frac{10}{9}x^{-\frac{4}{3}}+\frac{10}{9}x^{-\frac{4}{3}}\] Now, evaluate the second derivative at the critical points x = -2 and x = 0: \[f''(-2) = -\frac{10}{9}(-2)^{-\frac{4}{3}} + \frac{10}{9}(-2)^{-\frac{4}{3}} = \frac{40}{27} > 0\] Since \(f''(-2) > 0\), we have a relative minimum at x = -2. However, the second derivative is not defined at x = 0, so the second derivative test is inconclusive about the nature of the critical point at x = 0. We will use the first derivative test to classify the critical point at x = 0. Analyzing the sign of \(f'(x)\) around \(x = 0\): For \(-2 < x < 0\), both terms in \(f'(x)\) are negative, so \(f'(x) < 0\). This means the function is decreasing. For \(0 < x < 2\), the first term in \(f'(x)\) is positive and the second term is negative, so \(f'(x) > 0\). This means the function is increasing. Therefore, x = 0 is an inflection point.

Using the critical points from Step 2, we have: Increasing intervals: \((0, \infty)\) Decreasing intervals: \((-2, 0)\)

Using the information from the previous steps, we can sketch a graph of the function \[f(x) = x^{\frac{5}{3}} + 5x^{\frac{2}{3}}\]: 1. There is a relative minimum at x = -2. 2. There is an inflection point at x = 0. 3. The function is decreasing on the interval \((-2, 0)\). 4. The function is increasing on the interval \((0, \infty)\). The graph should display a curve that has a relative minimum at x = -2 and changes concavity at x = 0. The curve should decrease on the interval \((-2, 0)\) and increase on the interval \((0, \infty)\).