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Determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.

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Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.

asked Jan 29, 2015 in CALCULUS by anonymous

2 Answers

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Step 1:

The function is .

Consider .

Derivative on each side by .

Apply the power rule of derivative :.

Step 2:

To examine the behavior of a function, equate the derivative to zero.

The x values are .

answered Jan 31, 2015 by james Pupil
edited Jan 31, 2015 by yamin_math
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Step 3:

The function is .

The domain of the function is image.

There are four regions to examine the behavior of the function.

First region  image.

Consider a test point in the region.

The derivative is negative, the function is decreasing over image.

Second region  image.

Consider a test point image in the region.

The derivative is positive, the function is increasing over image.

Third region  image.

Consider a test point image in the region.

image

The derivative is negative, the function is decreasing over image.

Fourth region  image.

Consider a test point image in the region.

image

The derivative is positive, the function is increasing over image.

A monotonic function is increasing over image and image.

A monotonic function is decreasing over image and image.

Therefore the function is not strictly monotonic.

Solution :

The function is not strictly monotonic.

answered Jan 31, 2015 by james Pupil
edited Jan 31, 2015 by james

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