Calculus AB: Applications of the Derivative

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

Solution for Problem 1 >>

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,)

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

Solution for Problem 2 >>

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:

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Problem : Classify the critical points of f (x) = x ³ -3x ² - 9x .

Solution for Problem 3 >>

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.

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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.

Solution for Problem 4 >>

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.

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