10. $[-/ 1$ Points] DETAILS SCALC9 3.3.025. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Suppose the derivative of a function $f$ is \[ f^{\prime}(x)=(x-5)^{6}(x+8)^{9}(x-4)^{8} \] On what interval(s) is $f$ increasing? (Enter your answer using interval notation.) Need Help? Read It Submit Answer

Step 1 ：Given the derivative of a function $f$ is $f^{\prime}(x)=(x-5)^{6}(x+8)^{9}(x-4)^{8}$, we need to find the interval(s) where $f$ is increasing.

Step 2 ：The function $f$ is increasing where its derivative $f'(x)$ is positive. The derivative is a product of three terms, each of which is a power of a linear function. Each term is zero at a specific value of $x$, and changes sign at that value.

Step 3 ：Therefore, we need to find the values of $x$ where each term is zero, and then test the sign of $f'(x)$ in each interval determined by these values.

Step 4 ：The roots of the derivative are $x = -8, 4, 5$.

Step 5 ：The intervals where the derivative is positive, and therefore the function is increasing, are $(-8, 4)$ and $(4, 5)$.

Step 6 ：However, we need to consider the multiplicity of the roots. The function changes sign at a root if the root has an odd multiplicity. In this case, all roots have even multiplicity, so the function does not change sign at the roots.

Step 7 ：Therefore, the function is increasing on the entire real line.

Step 8 ：Final Answer: The function $f$ is increasing on the interval \(\boxed{(-\infty, \infty)}\).