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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 + x − 1 on [0, 4] ?
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Solution
We have,
Since polynomial function is everywhere continuous and differentiable.
Therefore, \[f\left( x \right)\] is continuous on \[\left[ 0, 4 \right]\] and differentiable on \[\left( 0, 4 \right)\]
Consequently, there exists some \[c \in \left( 0, 4 \right)\] such that
\[\Rightarrow 2x + 1 = \frac{20}{4}\]
\[ \Rightarrow 2x + 1 = 5\]
\[ \Rightarrow 2x = 4 \]
\[ \Rightarrow x = 2\]
Thus, \[c = 2 \in \left( 0, 4 \right)\] such that \[f'\left( c \right) = \frac{f\left( 4 \right) - f\left( 0 \right)}{4 - 0}\].
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