9. [-/11.12 Points] DETAILS MY NOTES SCALCET9 4.1.059. Find the absolute maximum and absolute minimum values of \( f \) on the given \[ f(t)=t-\sqrt[3]{t},[-1,4] \] absolute minimum value absolute maximum value

Alright, I need to find the absolute maximum and minimum values of the function \( f(t) = t - \sqrt[3]{t} \) on the interval \([-1, 4]\). Let's break this down step by step. First, I'll recall that to find the absolute extrema of a function on a closed interval, I need to evaluate the function at the critical points within the interval and at the endpoints of the interval. The largest and smallest of these values will be the absolute maximum and minimum, respectively. So, let's start by finding the critical points of \( f(t) \). Critical points occur where the derivative of the function is zero or undefined. Given \( f(t) = t - \sqrt[3]{t} \), I'll first find its derivative \( f'(t) \). The derivative of \( t \) with respect to \( t \) is 1. The derivative of \( \sqrt[3]{t} \) with respect to \( t \) is \( \frac{1}{3}t^{-\frac{2}{3}} \). So, \( f'(t) = 1 - \frac{1}{3}t^{-\frac{2}{3}} \). To find the critical points, set \( f'(t) = 0 \): \[ 1 - \frac{1}{3}t^{-\frac{2}{3}} = 0 \] Solving for \( t \): \[ \frac{1}{3}t^{-\frac{2}{3}} = 1 \] \[ t^{-\frac{2}{3}} = 3 \] Taking both sides to the power of \(-\frac{3}{2}\): \[ t = 3^{-\frac{3}{2}} \] \[ t = \frac{1}{3^{\frac{3}{2}}} \] \[ t = \frac{1}{3\sqrt{3}} \] \[ t \approx 0.1925 \] So, the critical point is at \( t \approx 0.1925 \). Next, I'll evaluate \( f(t) \) at the critical point and at the endpoints of the interval \([-1, 4]\). First, at \( t = -1 \): \[ f(-1) = -1 - \sqrt[3]{-1} \] \[ \sqrt[3]{-1} = -1 \] \[ f(-1) = -1 - (-1) = 0 \] Next, at \( t = 0.1925 \): \[ f(0.1925) = 0.1925 - \sqrt[3]{0.1925} \] Calculating \( \sqrt[3]{0.1925} \): \[ \sqrt[3]{0.1925} \approx 0.577 \] \[ f(0.1925) \approx 0.1925 - 0.577 = -0.3845 \] Finally, at \( t = 4 \): \[ f(4) = 4 - \sqrt[3]{4} \] \[ \sqrt[3]{4} \approx 1.587 \] \[ f(4) \approx 4 - 1.587 = 2.413 \] Now, comparing the values: - \( f(-1) = 0 \) - \( f(0.1925) \approx -0.3845 \) - \( f(4) \approx 2.413 \) From these calculations, the absolute maximum value of \( f(t) \) on the interval \([-1, 4]\) is approximately **2.413** at \( t = 4 \), and the absolute minimum value is approximately **-0.3845** at \( t \approx 0.1925 \).