Verify Lagrange's mean value theorem for the following function o… - Vidyadip

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.
$f(x) = x^2 + x - 1$ on $[0, 4]$

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Answer

Here, $f(x) = x^2 + x - 1$ on $[0, 4]$
f(x) is polynomial, so it is continuous is $[0, 4]$ and differentiable in $(0, 4)$
as every polynomial is continuous and differentiable everywhere. So, Lagrange's mean value theorem is applicable, so there exist a point $\text{c}\in[0,4]$ such that
$\text{f}'(\text{c})=\frac{\text{f}(4)-\text{f}(0)}{4-0}$
$\Rightarrow2\text{c}+1=\frac{\big((4)^2+4-1\big)-(0-1)}{4}$
$\Rightarrow2\text{c}+1=\frac{19+1}{4}$
$\Rightarrow2\text{c}+1=5$
$\Rightarrow\text{c}=2\in(0,4)$
Hence, Lagrange's mean value theorem is verified.

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