Verify Rolle's theorem for each of the following functions:
$f (x) = cos 2 x$ in $[0, π]$

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Verify Rolle's theorem for each of the following functions:

f (x) = cos 2 x

[0, π]

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Text solutionVerified

conditions of rolle's theorem

1.f(x) is continuous

2. f(x) is derivable

3.f(a) =f(b)

f(x) = cos 2x

Cosine function is continuous and differentiable at each point.
∴∴ f(x) is continous in the interval [0,π][0,π] and differentiable in the interval ]0,π]]0,π
Here
f(0) = cos 0 = 1
and f(π)=cos2π=1f(π)=cos2π=1
∴ f(0)=f(1)∴ f(0)=f(1)
Thus the conditions of Rolle's theorem satisfies .
Now let f'(c) = 0
⇒ −2sin2c=0 [∵f'(x)=−2sin2x]⇒ -2sin2c=0 [∵f′(x)=-2sin2x]
⇒ sin2c=0⇒ sin2c=0
⇒ 2c=0,π,2π....⇒ 2c=0,π,2π....
⇒ c=0,π/2,π....⇒ c=0,π/2,π....
c=π/2∈]0.π[c=π/2∈]0.π[
Hence Rolle's theorem verified .

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