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 \( f^{\prime \prime} \). \[ f(x)=4 x-5 x^{5 / 6} \] \( f^{\prime}(x)=\square \) \( f^{\prime \prime}(x)=\square \)

\[ \textbf{Step 1. Differentiate } f(x)=4x-5x^{\frac{5}{6}}. \] \[ f'(x)=\frac{d}{dx}\left(4x\right)-\frac{d}{dx}\left(5x^{\frac{5}{6}}\right)=4-5\cdot\frac{5}{6}x^{\frac{5}{6}-1}. \] \[ \text{Since } \frac{5}{6}-1=-\frac{1}{6}, \quad f'(x)=4-\frac{25}{6}x^{-\frac{1}{6}}. \] \[ \textbf{Step 2. Differentiate } f'(x)=4-\frac{25}{6}x^{-\frac{1}{6}}. \] \[ f''(x)=\frac{d}{dx}\left(4\right)-\frac{25}{6}\cdot\frac{d}{dx}\left(x^{-\frac{1}{6}}\right)=0-\frac{25}{6}\cdot\left(-\frac{1}{6}x^{-\frac{1}{6}-1}\right). \] \[ f''(x)=\frac{25}{36}x^{-\frac{7}{6}}. \] \[ \boxed{ \begin{aligned} f'(x)&=4-\frac{25}{6}x^{-\frac{1}{6}},\\[1mm] f''(x)&=\frac{25}{36}x^{-\frac{7}{6}}. \end{aligned} } \] \[ \textbf{Checking the results:} \] - \(\mathbf{f(x)=4x-5x^{\frac{5}{6}}}\) shows a linear term and a fractional power term. For large \(x\), the linear term \(4x\) dominates so \(f(x)\) increases roughly linearly. - \(\mathbf{f'(x)=4-\frac{25}{6}x^{-\frac{1}{6}}}\) is negative for very small \(x\) (since \(x^{-\frac{1}{6}}\) becomes very large), but as \(x\) increases, the term \(\frac{25}{6}x^{-\frac{1}{6}}\) decreases and \(f'(x)\) tends to \(4\). Thus, its graph transitions from steeply negative to approaching the value \(4\). - \(\mathbf{f''(x)=\frac{25}{36}x^{-\frac{7}{6}}}\) is positive for \(x>0\) indicating that \(f'(x)\) is increasing and the graph of \(f(x)\) is concave upward for \(x>0\). These observations are consistent with the behavior of the graphs of \(f\), \(f'\), and \(f''\).