Increasing/Decreasing composites functions

  • Thread starter jordanfc
  • Start date
  • Tags
    Functions
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)).
  • #1
jordanfc
4
0
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?
 
Last edited:
Physics news on Phys.org
  • #2
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).
 

FAQ: Increasing/Decreasing composites functions

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.

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.

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.

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.

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.

Similar threads

Is the function v(r) = (e^(-mr))/r a solution to the Yukawa PDE equation?
Replies
1
Views
1K
Is d'Alembert's Formula Correct for Neumann Boundary Conditions in PDEs?
Replies
4
Views
2K
Applying a substitution to a PDE
Replies
4
Views
1K
Given unit impulse response, obtain differential equation
Replies
7
Views
1K
Is It Possible to Invert a Homotopy?
Replies
5
Views
713
The integral of a function ##f(x)## from its graph
Replies
15
Views
2K
PDE and the separation of variables
Replies
11
Views
1K
How should I find ## x_{2}(t) ## for this nonlinear integro-differential equation?
Replies
6
Views
406
Prove that a locally constant function is constant on a connected X
Replies
20
Views
3K
Finding where this function is increasing or decreasing
Replies
10
Views
1K

Hot Threads

Recent Insights

Back
Top