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

Explanation

Given, on `[a, b], f'(x) gt 0`
`:. ` f is a differentiable function on [a, b] and every differentiable function is continuous, so f is continuous on [ a, b].
Let` x_(1), x_(2) in [a, b] and x_(2) gt x_(1)`, then from Lagrange's theorem, there exists ` c in [ a, b]` such that
`f'(c)= (f(x_(2))-f(x_(1)))/(x_(2)-x_(1))`
` rArrf(x_(2))-f(x_(1))=(x_(2)-x_(1))f(c)`
` rArr f(x_(2))-f(x_(1)) gt 0 " "(.:' x_(2) gt x_(1))`
`and f'(x) gt 0 rArr f(x_(2)) gt f(x_(1))`
` :. "For "x_(1) lt x_(2) rArr f(x_(1))lt f(x_(2))`
Therefore, f in an inceasing function on (a, b).