Increasing/Decreasing composites functions

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In summary, for f(x) to be increasing, f'(x) must be greater than 0, which can be achieved when u(x) is positive. However, for g(x) to be increasing, u'(x) must be negative and the chain rule must be applied to analyze g'(x). Therefore, there may be values of x where g(x) is not increasing due to the chain rule and the bounded value of 2u'(u(x)).
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jordanfc
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Given that u(x) is always positive and u'(x) < 0, I need to find values of x so that f(x) and g(x) are increasing. f(x) = [u(x)]^2 and g(x) = u(u(x)).

for f(x) is increasing when f'(x) > 0. so f'(x) = 2u(x) => 2u(x) > 0. would f(x) always be increasing since 2u(x) will always be increasing ( u(x) is always positive, 2u(x) will always be positive as well)?

and for g'(x) > 0 => [u(u(x))]' => u'(u(x)*(u(x))' = 2u'(u(x))...set larger than 0...since u'(x) is always negative and 2u'(u(x)) is bounded by 2u', does that inequality not hold and so g(x) is never increasing?
 
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f'(x) is not 2u(x), you need to apply the chain rule. You must also use the chain rule when you analyze g'(x).
 

Related to Increasing/Decreasing composites functions

1. How do I determine if a composite function is increasing or decreasing?

In order to determine if a composite function is increasing or decreasing, you need to examine the individual functions that make up the composite function. If the inner function is increasing and the outer function is also increasing, then the composite function will be increasing. If the inner function is decreasing and the outer function is also decreasing, then the composite function will be decreasing. If the inner function and outer function have opposite trends (one is increasing while the other is decreasing), then the composite function will be decreasing.

2. Can a composite function be both increasing and decreasing?

No, a composite function cannot be both increasing and decreasing at the same time. The overall trend of a composite function is determined by the trend of its individual functions. If the inner and outer functions have opposite trends, then the composite function will be decreasing. If the inner and outer functions have the same trend, then the composite function will be either increasing or decreasing.

3. How can I use calculus to analyze increasing/decreasing composite functions?

Calculus can be used to analyze increasing/decreasing composite functions by taking the derivative of the composite function. The derivative will tell you the rate of change of the composite function at a given point. If the derivative is positive, then the composite function is increasing. If the derivative is negative, then the composite function is decreasing. You can also find critical points and use the first or second derivative test to determine if the function is increasing or decreasing at those points.

4. Can a composite function have more than one increasing/decreasing interval?

Yes, a composite function can have multiple increasing/decreasing intervals. This can occur when the inner and outer functions have different trends in different intervals. For example, the inner function could be increasing in one interval while the outer function is decreasing, resulting in a decreasing composite function. It is important to analyze the trends of each individual function within the composite function to determine the overall trend.

5. Are there any special cases for determining the increasing/decreasing nature of a composite function?

One special case for determining the increasing/decreasing nature of a composite function is when the inner function is a constant. In this case, the composite function will either be increasing or decreasing based on the trend of the outer function. Another special case is when the outer function is a constant. In this case, the composite function will have the same trend as the inner function. It is also important to consider any vertical asymptotes or points of discontinuity in the composite function, as these can affect its overall trend.

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