Question

Find all local maximums and minimums of each function.
f(x)=x³-3 x²-9 x-5

Answer 1

To find the local maximums and minimums, find the critical points and evaluate the function at these points and the endpoints of the given interval.

Step-by-step explanation:

To find the local maximums and minimums of the function f(x) = x³ - 3x² - 9x - 5, we first need to find the critical points. These are the points where the derivative of the function is equal to zero or undefined.

Using calculus, we find that the derivative of f(x) is f'(x) = 3x² - 6x - 9. Setting f'(x) equal to zero and solving for x, we get x = -1 and x = 3. These are the critical points.

Next, we evaluate the function f(x) at these critical points and the endpoints of the given interval to determine if they correspond to local maximums or minimums.

At x = -1, f(-1) = -18. At x = 3, f(3) = -11. The endpoints of the interval are x = 0 and x = 20. Evaluating the function at these points, we get f(0) = -5 and f(20) = 3155.

So, the local maximum is f(0) = -5 and the local minimums are f(-1) = -18 and f(20) = 3155.