RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 11 of Exercise 17.2

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11

Show that f(x) = cos2 x is a decreasing function on (0, π/2).

Given:- Function f(x) = cos2 x


Theorem:- Let f be a differentiable real function defined on an open interval (a,b).


(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)


(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)


Algorithm:-


(i) Obtain the function and put it equal to f(x)


(ii) Find f’(x)


(iii) Put f’(x) > 0 and solve this inequation.


For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.


Here we have,


f(x) = cos2 x


⇒


⇒ f’(x) = 3cosx(–sinx)


⇒ f’(x) = –2sin(x)cos(x)


⇒ f’(x) = –sin2x ; as sin2A = 2sinA cosA


Now, as given



⇒ 2x ∈ (0,π)


⇒ Sin(2x)> 0


⇒ –Sin(2x)< 0


⇒ f’(x) < 0


hence, Condition for f(x) to be decreasing


Thus f(x) is decreasing on interval


Hence proved


Chapter Exercises

Exercise 17.1
Exercise 17.2
MCQ

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