Q. 665.0( 3 Votes )

Verify the Rolle’s theorem for each of the functions


Answer :

Given: f(x) = sin4x + cos4x

Now, we have to show that f(x) verify the Rolle’s Theorem


First of all, Conditions of Rolle’s theorem are:


a) f(x) is continuous at (a,b)


b) f(x) is derivable at (a,b)


c) f(a) = f(b)


If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0


Condition 1:


f(x) = sin4x + cos4x


Since, f(x) is a trigonometric function and trigonometric function is continuous everywhere


f(x) = sin4x + cos4x is continuous at


Hence, condition 1 is satisfied.


Condition 2:


f(x) = sin4x + cos4x


On differentiating above with respect to x, we get


f’(x) = 4 × sin3 (x) × cos x + 4 × cos3 x × (- sin x)



f’(x) = 4sin3 x cos x – 4 cos3 x sinx


f’(x) = 4sin x cos x [sin2x – cos2 x]


f’(x) = 2 sin2x [sin2x – cos2 x]


[ 2 sin x cos x = sin 2x]


f’(x) = 2 sin 2x [- cos 2x]


[ cos2 x – sin2 x = cos 2x]


f’(x) = - 2 sin 2x cos 2x


f(x) is differentiable at


Hence, condition 2 is satisfied.


Condition 3:


f(x) = sin4x + cos4x


f(0) = sin4(0) + cos4(0) = 1




Hence, condition 3 is also satisfied.


Now, let us show that c () such that f’(c) = 0


f(x) = sin4x + cos4x


f’(x) = - 2 sin 2x cos 2x


Put x = c in above equation, we get


f’(c) = - 2 sin 2c cos 2c


, all the three conditions of Rolle’s theorem are satisfied


f’(c) = 0


- 2 sin 2c cos 2c = 0


sin 2c cos 2c = 0


sin 2c = 0


2c = 0


c = 0


Now, cos 2c = 0






Thus, Rolle’s theorem is verified.


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