1. **Problem Statement:** Find the absolute maximum value of the function $f(x)$ on the interval $[-4,6]$ given the derivative $f'(x)$ and some key points. 2. **Recall:** The absolute maximum of $f(x)$ on a closed interval occurs at critical points where $f'(x)=0$ or at the endpoints of the interval. 3. **Given candidates for $x$ where $f(x)$ might have extrema:** $x=-4$, $x=0$, and $x=6$. 4. **Values of $f(x)$ at these points are given or derived:** - At $x=-4$, $f(x) = -2$ - At $x=0$, $f(x) = -11$ - At $x=6$, $f(x) = -4 - 2n$ 5. **Compare these values:** - $-2$ at $x=-4$ - $-11$ at $x=0$ - $-4 - 2n$ at $x=6$ 6. Since $-2 > -11$ and $-2 > -4 - 2n$ (assuming $n$ is a positive number or zero), the maximum value is $-2$ at $x=-4$. 7. **Final answer:** $$\boxed{\text{The absolute maximum value of } f(x) \text{ on } [-4,6] \text{ is } -2 \text{ at } x=-4.}$$