Problem : f (x) = x3 -4x2 - 4. Show where f (x) is increasing and where it is decreasing.

f'(x) = 3x2 - 8x;
f'(x) = 0 at x = 0 and x = . The sign of the derivative is reported on the number line below: So, f is increasing on (- ∞, 0) and (,∞), and it is decreasing on (0,)

Problem : f (x) = sin(x). Where is f increasing and decreasing on the interval [0, 2Π]?

f'(x) = - cos(x);
f'(x) = 0 at x = and x = .
The sign of the derivative and the behavior of the function is indicated below:

Problem : Classify the critical points of f (x) = x3 -3x2 - 9x.

f'(x) = 3x2 - 6x - 9 The sign of the derivative is shown in the figure below. Because the derivative is positive to the left of x = - 1 and negative to the right, x = - 1 is a local maximum; x = 3 is a local minimum.

Problem : Below is the graph of the derivative of a function f. Find regions where f is increasing and decreasing, and classify the critical points.

The sign of f'(x) and the behavior of f is depicted below: While b, d, f, and h are critical points, only d and h are local extrema. d is a local maximum and h is a local minimum.