Let A be a set, and let f, g: A be functions. Suppose that f and g are increasin

Question

Let A be a set, and let f, g: A be functions. Suppose that f and g are increasing.

Is f – g necessarily either increasing or decreasing? Give a proof or a counterexample.

f and g are increasing function

f’(x) 0, g’(x) 0 for all x A

If (f-g)’ = (f’ – g’)(x)

f’(x) – g’(x) 0

if f’(x) g’(x)

Then (f-g)(x) is increasing

if f’(x) < g’(x)

Then (f-g)’(x) < 0

(f-g)(x) is decreasing function

I need help finding examples of f and g increasing and f - g increasing and then another example of f and g increasing but f - g decreasing.

Explanation / Answer

f(x) = 2x + 4

g(x) = x

f'(x) = 2

g'(x) = 1

f'(x) - g'(x) = 1 > 0

hence

f-g in increasing

now simply exchange f and g

g(x) = 2x + 4

f(x) = x

f'(x) = 1

g'(x) = 2

f'(x) - g'(x) = -1 <0

hence

f-g is decreasing

hence the statement " f – g necessarily either increasing or decreasing" is false

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