Question

Verify Rolle's theorem for the function f(x)=sinx+cosx+7 where x belongs to [0,2π]​

Answer 1

Step-by-step explanation:

Since sine and cosine bring continuous lies in interval [0,2π]

f(x)=sinx+cosx−1

f(0)=0+1−1

f(0)=0

f(2π)=1+0−1

f(2π)=0

f(0)=f(2π)

∴ Rolle's theorem are satisfied

f′(x)=cosx−sinx

So, there must exist some c∈(0,2π) such

that f′(c)=0

f′(c)=cosc−sinc

cosc−sinc=0

cosc=sinc

∴ c=4π

Thus c=4π∈(0,2π)

∴ Rolle's theorem is verified