Let f be a function defined on [a, b] such that f ′(x) 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Question

Let f be a function defined on [a, b] such that f ′(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Philoid · Accepted Answer

Since, f’(x) > 0 on (a,b)

Then, f is a differentiating function (a,b)

Also, every differentiable function is continuous,

Therefore, f is continuous on [a,b]

Let x1, x2 ϵ (a,b) and x2 > x1 then by LMV theorem, there exists c ϵ (a,b) s.t.

f’(c) =

Therefore, f is an increasing function.