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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). - CBSE (Commerce) Class 12 - Mathematics

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Question

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

Solution

Let x1, x2∈(a,b) such that x1<x2.
Consider the sub-interval [x1, x2]. Since f (x) is differentiable on (a, b) and [x1, x2]⊂(a,b).
Therefore, f(x) is continous on [x1, x2] and differentiable on (x1, x2).

By the Lagrange's mean value theorm, there exists c∈(x1, x2) such that

f'(c)=f(x2)-f(x1)x2-x1          ...(1)

Since f'(x) > 0 for all x∈(a,b), so in particular, f'(c) > 0

f'(c)>0⇒f(x2)-f(x1)x2-x1>0          [Using (1)]      

⇒f(x2)-f(x1)>0          [∵ x2-x1>0 when x1<x2]
⇒f(x2)>f(x1)⇒f(x1)<f(x2)

Since x1, x2 are arbitrary points in (a,b).

Therefore, x1<x2⇒f(x1)<f(x2) for all x1,x2∈(a, b)
Hence, f (x) is increasing on (a,b).

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APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 6: Application of Derivatives
Q: 16 | Page no. 243
Solution 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). Concept: Increasing and Decreasing Functions.
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