Increasing and Decreasing Functions?

Question

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?

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Answer 1 · 2017-07-11T18:06:11

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#?