Questions: Using the sign chart for f'(x)=-3(x-8)(x+5)^2, and only that sign chart, what conclusions can be drawn about the graph of f when x=-5?

Solution Steps

The given first derivative is $f'(x) = -3(x-8)(x+5)^2$. We want to determine the behavior of the graph of f at x = -5.

When x is slightly less than -5, (x+5) is negative, so $(x+5)^2$ is positive, and (x-8) is negative. Therefore, $f'(x) = -3 * (negative) * (positive)$, making $f'(x)$ positive.

When x is slightly greater than -5, (x+5) is positive, so $(x+5)^2$ is positive, and (x-8) is negative. Therefore, $f'(x) = -3 * (negative) * (positive)$, making $f'(x)$ positive.

Since the sign of $f'(x)$ does not change around x = -5, there is a horizontal tangent but no relative extrema at x = -5. The derivative is zero at x = -5, so there is a horizontal tangent.

Final Answer: