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CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
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. $f(x) = x^2 - 2x + 4$ on $[1, 5]$
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Answer

We have,$f(x) = x^2 - 2x + 4$
Since a polynomial function is everywhere continuous and differentiable.
Therefore, $f(x)$ is continuous on $[1, 5]$ and differentiable on $(1, 5).$
Thus, both conditions of Lagrange's mean value theorem are satisfied.
So, there must exist at least one real number $\text{c}\in(1,5)$ such that
$\text{f}'(\text{c})=\frac{\text{f}(5)-\text{f}(-1)}{5-1}=\frac{\text{f}(5)-\text{f}(-1)}{4}$
Now,$ f(x) = x^2 - 2x + 4$
$\Rightarrow f'(x) = 2x - 2$
$\Rightarrow f(5) = 25 - 10 + 4 = 19$
$\Rightarrow f(1) = 1 - 2 + 4 = 3$
$\therefore\ \text{f}'(\text{x})=\frac{\text{f}(5)-\text{f}(-1)}{4}$
$\Rightarrow2\text{x}-2=\frac{19-3}{4}$
$\Rightarrow2\text{x}-2-4=0$
$\Rightarrow\text{x}=\frac{6}{2}=3$
Thus, $\text{c}=3\in(1,5)$ such that $\text{f}'(\text{c})=\frac{\text{f}(5)-\text{f}(-1)}{5-1}$
Hence, Lagrange's mean value theorem is verified.

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