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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) = x2 − 2x + 4 on [1, 5] ?
Solution
We have,
\[f\left( x \right) = x^2 - 2x + 4\]
Since a polynomial function is everywhere continuous and differentiable.
Therefore, \[f\left( x \right)\] is continuous on \[\left[ 1, 5 \right]\] and differentiable on \[\left( 1, 5 \right)\] .
So, there must exist at least one real number
\[\Rightarrow 2x - 2 = \frac{19 - 3}{4}\]
\[ \Rightarrow 2x - 2 - 4 = 0\]
\[ \Rightarrow x = \frac{6}{2} = 3\]
Thus, \[c = 3 \in \left( 1, 5 \right)\] such that
\[f'\left( c \right) = \frac{f\left( 5 \right) - f\left( 1 \right)}{5 - 1}\] .
Hence, Lagrange's theorem is verified.
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