Question

$\bf \: Verify \: the \: Rolle’s \: Theorem \: for \: functions \: on \: the \: indicated \: intervals: \\ \sf \implies f(x) = cos \: 2x \: in \: the \: interval \: [0, \pi]$

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Answer 1

$\large\underline{\sf{Solution-}}$

Given function,

$\rm :\longmapsto\:f(x) = cos2x \: \: \: in \: [0, \: \pi]$

Step :- 1 Continuity

We know, cosine function is always continuous for every real number.

$\rm\implies \:\:f(x) = cos2x \: is \: continuous \: on \: [0, \: \pi]$

Step :- 2 Differentiability

$\rm :\longmapsto\:f(x) = cos2x$

So,

$\rm :\longmapsto\:f'(x) = - sin2x$

Hence,

$\rm\implies \:\:f(x) = cos2x \: is \: differentiable \: on \: (0, \: \pi)$

Step :- 3

$\rm :\longmapsto\:f(0) = cos0 = 1$

and

$\rm :\longmapsto\:f(\pi) = cos2\pi = 1$

So,

$\rm\implies \:f(0) = f(\pi)$

So, it means, Rolle's Theorem is applicable.

It implies, there exist atleast one real number c ∈ (0, π)

such that

$\rm :\longmapsto\:f'(c) = 0$

$\rm :\longmapsto\: - sin2c = 0$

$\rm :\longmapsto\: sin2c = 0$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ sinx = 0 \: \rm\implies \: \sf x = n\pi \: \forall \: n \in \: Z}}}$

So, using this identity, we get

$\rm :\longmapsto\: 2c =\: n\pi \: \forall \: n \in \: Z$

$\rm :\longmapsto\: c =\:\dfrac{n\pi}{2} \: \forall \: n \in \: Z$

$\rm\implies \:c = \dfrac{\pi}{2}\: \: \in \: (0, \: \pi)$

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Definition of Rolle's Theorem

Let f(x) be a real valued function defined on [a, b] such that

(i) f(x) is continuous on [a, b]

(ii) f(x) is differentiable on (a, b)

(iii) f(a) = f(b)

then, there exist atleast one real number c x∈ (a, b) such that f'(c) = 0.

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More about T - Formula

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf T-eq & \bf Solution \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf sinx = 0 & \sf x = n\pi \: \forall \: n \in \: Z\\ \\ \sf cosx = 0 & \sf x = (2n + 1)\dfrac{\pi}{2}\: \forall \: n \in \: Z\\ \\ \sf tanx = 0 & \sf x = n\pi\: \forall \: n \in \: Z\\ \\ \sf sinx = siny & \sf x = n\pi + {( - 1)}^{n}y \: \forall \: n \in \: Z\\ \\ \sf cosx = cosy & \sf x = 2n\pi \pm \: y\: \forall \: n \in \: Z\\ \\ \sf tanx = tany & \sf x = n\pi + y \: \forall \: n \in \: Z\end{array}} \\ \end{gathered}\end{gathered}$

Answer 2

[Refer to the above attachment]

Hope you have satisfied. ⚘