Question

verify lmvt for (x-1)/(x-3) on [ 4,5 ]​

Answer 1

$\textbf{Given:}$

$f(x)=(x-1)(x-3)$

$\textbf{To verify:}$

$\text{Lagrange's mean value theorem}$

$\textbf{Solution:}$

$f(x)=x^2-4x+3$

$f'(x)=2x-4=2(x-2)$

$\text{Since f(x) is a polynomial,}$

$\text{f(x) is continuous on [4,5]}$

$\text{f(x) is differentiaable in (4,5)}$

$\therefore\text{Conditions of lagrange's mean value theorem are satisfied}$

$\text{Hence, by lagrange's mean value theorem,}$

$\text{there exists a c \in (4,5) such that f'(c)=\dfrac{f(5)-f(4)}{5-4} }$

$\text{Now,}$

$f'(c)=2(c-2)$

$f(5)=5^2-4(5)+3=25-20+3=8$

$f(4)=4^2-4(4)+3=16-16+3=3$

$f'(c)=\dfrac{f(5)-f(4)}{5-4}$

$\implies\,2(c-2)=\dfrac{8-3}{1}$

$\implies\,2(c-2)=5$

$\implies\,c-2=\frac{5}{2}$

$\implies\,c=\frac{5}{2}+2$

$\implies\bf\,c=\frac{9}{2}\,{\in}\,(4,5)$

$\textbf{Hence, Lagrange's mean value theorem verified}$

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Answer 2

Answer:

Step-by-step explanation: