Monotonically increasing/decreasing functions

whenever a function is expressed in terms of another function,how do we find whether the function is increasing or decreasing?

say ,for example ,we have :

h(x)= f(2x)+f(x)

pls tell anyone,what procedure we need to follow to find whether the above given function is increasing/monotonically increasing or decreasing/monotonically decreasing.thanks !
 

gb7nash

Homework Helper
802
1
There are two ways that I can think of. Assuming that the function is continuously differentiable, take the derivative and set equal to 0. Make a number line and test a point between every 0 to see if it's increasing or decreasing. Also, if there is no zero, then you have a monotonic increasing/decreasing function automatically.

Assuming your function is h(x), the other way is to consider h(x+a) - h(x), where a is an arbitrary positive number. If you can show that h(x+a) - h(x) is positive, then h is monotonically increasing.
 
Last edited:
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To show that [itex]f[/itex] is monotonically increasing, we need to show that for any [itex]\Delta{x} > 0[/itex], [itex]f(x + \Delta{x}) > f(x)[/itex] for all [itex]x[/itex] in the domain; or equivalently, [itex]f(x + \Delta{x}) - f(x) > 0[/itex]. An equivalent definition is that [itex]f(x_1) < f(x_2)[/itex] for all [itex]x_1, x_2[/itex] in the domain of [itex]f[/itex] with [itex]x_1 < x_2[/itex].

For your problem, try to evaluate [itex]h(x + \Delta{x}) - h(x)[/itex] with the assumption [itex]f(x + \Delta{x}) > f(x)[/itex].
But I think you need more information about [itex]f(x)[/itex] to solve this.
 

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