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Problems from "Using the First Derivative to Analyze Functions"

Problems from "Using the First Derivative to Analyze Functions"

Problems from "Using the First Derivative to Analyze Functions"

Problems from "Using the First Derivative to Analyze Functions"

Problems from "Using the First Derivative to Analyze Functions"

Problems from "Using the First Derivative to Analyze Functions"

Problem : f (x) = x 3 -4x 2 - 4 . Show where f (x) is increasing and where it is decreasing.

f'(x) = 3x 2 - 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) = x 3 -3x 2 - 9x .

f'(x) = 3x 2 - 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.