Monotonically increasing/decreasing functions

[gliteringstar]

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]

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.

 

[Dr. Seafood]

To show that $f$ is monotonically increasing, we need to show that for any $\Delta{x} > 0$, $f(x + \Delta{x}) > f(x)$ for all $x$ in the domain; or equivalently, $f(x + \Delta{x}) - f(x) > 0$. An equivalent definition is that $f(x_1) < f(x_2)$ for all $x_1, x_2$ in the domain of $f$ with $x_1 < x_2$.

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