CBSE (Science) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution for Verify Lagrange'S Mean Value Theorem for the Following Function on the Indicated Intervals. Find a Point 'C' in the Indicated Interval as Stated by the F(X) = Sin X − Sin 2x − X on [0, π] ? - CBSE (Science) Class 12 - Mathematics

Login
Create free account


      Forgot password?

Question

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = sin x − sin 2x − x on [0, π] ?

Solution

We have ,

\[f\left( x \right) = \sin x - \sin2x - x\]

Since

\[\sin x, \sin2x \text { & }x\] are everywhere continuous and differentiable]

Therefore, 

\[f\left( x \right)\] is continuous on \[\left[ 0, \pi \right]\] and differentiable on \[\left( 0, \pi \right)\]
Thus, both the conditions of lagrange's theorem are satisfied.
Consequently, there exists some 
\[c \in \left( 0, \pi \right)\]  such that
\[f'\left( c \right) = \frac{f\left( \pi \right) - f\left( 0 \right)}{\pi - 0} = \frac{f\left( \pi \right) - f\left( 0 \right)}{\pi}\]
Now,
\[f\left( x \right) = \sin x - \sin2x - x\]
\[f'\left( x \right) = \cos x - 2\cos2x - 1\],
\[f\left( \pi \right) = - \pi\],
\[f\left( 0 \right) = 0\]
∴ \[f'\left( x \right) = \frac{f\left( \pi \right) - f\left( 0 \right)}{\pi - 0}\]

\[\Rightarrow \cos x - 2\cos2x - 1 = - 1\]

\[ \Rightarrow \cos x - 2\cos2x = 0\]

\[ \Rightarrow \cos x - 4 \cos^2 x = - 2 \]

\[ \Rightarrow 4 \cos^2 x - \cos x - 2 = 0\]

\[ \Rightarrow \cos x = \frac{1}{8}\left( 1 \pm \sqrt{33} \right)\]

\[ \Rightarrow x = \cos^{- 1} \left[ \frac{1}{8}\left( 1 \pm \sqrt{33} \right) \right]\]

Thus, 

\[c = \cos^{- 1} \left( \frac{1 \pm \sqrt{33}}{8} \right) \in \left( 0, \pi \right)\] such that 

\[f'\left( c \right) = \frac{f\left( \pi \right) - f\left( 0 \right)}{\pi - 0}\].

Hence, Lagrange's theorem is verified.

  Is there an error in this question or solution?
Solution Verify Lagrange'S Mean Value Theorem for the Following Function on the Indicated Intervals. Find a Point 'C' in the Indicated Interval as Stated by the F(X) = Sin X − Sin 2x − X on [0, π] ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
S
View in app×