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).

Home » 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).

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).

Let I be the interval

Given: for all in an interval I. Let I with

By Lagrange’s Mean Value Theorem, we have,

where

Now

……….(i)

Also, for all in an interval I

From eq. (i),

Thus, for every pair of points I with

Therefore, is strictly increasing in I.

i

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