Question

Increasing and Decreasing Functions?

Determine if each statement is True or False. If it is true explain why is so, if it is false give a counterexample.

Is the sum of two increasing functions increasing?
Is the product of two increasing functions increasing?

Answer 1

Sum: Yes

In the proof I assume `f` and `g` are strictly increasing, but the proof will work for non-decreasing as well. Just change the `<` to `<=`.

Suppose `f` and `g` are increasing on interval `I`.

Then for any `x_1`, `x_2` in `I`, we have

`f(x_1) < f(x_2)` and `g(x_1) < g(x_2)`.

This entails that

`(f+g)(x_1) = f(x_1) + g(x_1) < f(x_2)+ g(x_2) = (f+g)(x_2)`

So `f+g` is increasing.

Product: Not necessarily.

Observe that the proof used above will not work for a product if one of the functions is negative. That's a key for finding a counter-example.

`f(x) = x^2` and `g(x) = -1/x` are both increasing on `(0,oo)`, but the product

`(fg)(x) = f(x)g(x) = -x` is decreasing on `(0,oo)`

Bonus

If `f` and `g` are both non-negative and increasing on `I`, then the product is increasing on `I`.

What is both are non-positive and increasing on `I`?