Question

Verify Rolle's theorem for f(x)= sin x-1 in [pi/2 , 5pi/2]

Answer 1

Answer:

f(x)=e

−x

sinx,x∈(0,π)

For Rolle's Theorem, f(0)=f(π) & f(x) must be continuous & differentiable over [0,π]

Let us check if f(0)=f(π)

⇒f(0)=e

−0

sin(0)=0

⇒f(π)=e

−π

sin(π)=0

Therefore, f(0)=f(π)

The function e

−x

& sinx are both continuous & differentiable over [0,π]

Therefore, Rolle's Theorem can be applied for the function given.

There exists

′

c

′

such that f

′

(c)=0

f

′

(x)=−e

−x

sinx+e

−x

cosx

f

′

(c)=−e

−c

sin(c)+e

−c

cos(c)=0⇒e

−c

[cosc−sinc]=0

⇒cosc=sinc⇒tanc=1⇒

c=

4

π

Hence, Rolle's Theorem is verified.