Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of \( f, f^{\prime} \), and \( f^{\prime \prime} \). \[ f(x)=4 x-5 x^{5 / 6} \] \( f^{\prime}(x)=\square \) \( f^{\prime \prime}(x)=\square \)

To find the first and second derivatives of the function \( f(x) = 4x - 5x^{5/6} \), we can use differentiation rules. First, we compute the first derivative \( f'(x) \): \[ f'(x) = \frac{d}{dx}(4x) - \frac{d}{dx}(5x^{5/6}) = 4 - 5 \cdot \frac{5}{6} x^{5/6 - 1} \] \[ f'(x) = 4 - \frac{25}{6}x^{-1/6} \] Next, we compute the second derivative \( f''(x) \): \[ f''(x) = \frac{d}{dx}\left(4 - \frac{25}{6}x^{-1/6}\right) = 0 + \frac{25}{6} \cdot \frac{1}{6} x^{-1/6 - 1} \] \[ f''(x) = \frac{25}{36}x^{-7/6} \] So the final results are: \[ f'(x) = 4 - \frac{25}{6}x^{-1/6} \] \[ f''(x) = \frac{25}{36}x^{-7/6} \] Now, to check if these derivatives make sense, you can graph \( f(x) \), \( f'(x) \), and \( f''(x) \) using graphing software or online graphing tools. You should note that the first derivative \( f'(x) \) generally indicates the slope of \( f(x) \), where positive values suggest rising and negative values suggest falling, while \( f''(x) \) gives insight into the concavity of the curve—whether it’s curving upwards or downwards!