RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 38 of Exercise 17.2

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38

Let F defined on [0, 1] be twice differentiable such that | f”(x) ≤ 1 for all x ϵ [0, 1]. If f(0) = f(1), then show that |f’(x) | < 1 for all x ϵ [0, 1] ?

As f(0) = f(1) and f is differentiable, hence by Rolles theorem:

for some c [0,1]


let us now apply LMVT (as function is twice differentiable) for point c and x [0,1],


hence,


f ”(d)


⇒ f ”(d)


⇒ f ”(d)


A given that | f ”(d)| <=1 for x [0,1]


⇒


⇒


Now both x and c lie in [0,1], hence [0,1]


Chapter Exercises

Exercise 17.1
Exercise 17.2
MCQ

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32

Prove that the function f given by f(x) = x3 – 3x2 + 4x is strictly increasing on R ?
33

33 Prove that the function f(x) = cos x is :

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38

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39

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