# 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).

**Solution:**

Increasing functions are those functions that increase monotonically within a particular domain,

and decreasing functions are those which decrease monotonically within a particular domain.

Let x_{1}, x_{2} ∈ (a, b) such that x_{1} > x_{2}

Consider the sub-interval [x_{1}, x_{2}]

Since f (x) is differentiable on (a, b) and [x_{1}, x_{2}] ⊂ (a, b).

Therefore,

f (x) is continuous on [x_{1}, x_{2}] and differentiable on (x_{1}, x_{2})

By the Lagrange's mean value theorem,

there exists c ∈ (x_{1}, x_{2}) such that

f (c) = (f (x_{2}) - f (x_{1}))/(x_{1} - x_{2}) .... (1)

Since, f' (x) > 0 for all x ∈ (a, b),

so in particular,

f' (c) > 0

⇒ [f (x_{2}) - f (x_{1})] / (x_{1} - x_{2}) > 0 [Using (1)]

⇒ f (x_{2}) - f (x_{1}) > 0

⇒ f (x_{2}) > f (x_{1})

⇒ f (x_{1}) < f (x_{2})

Since, x_{1}, x_{2 } are arbitrary points in (a, b).

Therefore, x_{1} < x_{2}

⇒ f (x_{1}) < f (x_{2}) for all x_{1}, x_{2} ∈ (a, b)

Hence, f (x) is increasing on (a, b)

NCERT Solutions Class 12 Maths - Chapter 6 Exercise ME Question 16

## 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)

**Summary:**

Given that f be a function defined on [a, b] such that f' (x) > 0 , for all x ∈ (a, b). Hence we have proved that f is an increasing function on (a, b)

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