10. $[-/ 1$ Points]
DETAILS SCALC9 3.3.025.
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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.)
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Final Answer: The function $f$ is increasing on the interval \(\boxed{(-\infty, \infty)}\).
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 \(\boxed{(-\infty, \infty)}\).