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) = 2x² - 3x + 1 on [1, 3]

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Answer

Here,

f(x) = 2x² - 3x + 1 on [1, 3]

We know that a polynomial function is continuous and differentiable.

So, f(x) is continuous in [1, 3] and f(x) differentiable in (1, 3).

So, Lagrange's mean value theorem is applicable.

So, there must exist at least one real number $\text{c}\in(1,3)$ such that

$\text{f}'(\text{c})=\frac{\text{f}(3)-\text{f}(-1)}{3-1}$

$\Rightarrow4\text{c}-3=\frac{(2(3)^2-3(3)+1)-(2-3+1)}{3-1}$

$\Rightarrow4\text{c}-3=\frac{10}{2}$

$\Rightarrow4\text{c}=5+3$

$\Rightarrow4\text{c}=8$

$\Rightarrow\text{c}=2\in(1,3)$

Hence, Lagrange's mean value theorem is verified.

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