Answer :

Final answer:

To determine where the function f(x) = -2x³ - 24x² + 54x - 19 is increasing, we compute the derivative, find the critical points, and then use a sign chart to identify where the derivative is positive, indicating where the function is increasing.

Explanation:

To find the intervals where the function f(x) = -2x³ - 24x² + 54x - 19 is increasing, we need to compute the derivative of the function, set it equal to zero, and solve for x. We then determine where the derivative is positive, indicating an increasing function.

The derivative of f(x) is f'(x) = -6x² - 48x + 54. Setting this equal to zero and solving for x gives us the critical points, these tell us the intervals we should check for increasing or decreasing trends.

We use a sign chart to examine the behavior of f'(x) to the left and right of each critical point. The intervals where f'(x) is positive are the intervals where f(x) is increasing. This applies our understanding of derivatives and how they relate to the increasing or decreasing behavior of a function.

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