Daily Question 106

Daily Question 106

$\textbf{Q106.}$ Let $f(x)=\ln x$ and $g(x)=x^2$. If $c\in (4,5)$ then $c\ln\left(\dfrac{4^{25}}{5^{16}}\right)$ equals to

(A) $c\ln 5 - 8$

(B) $2(c^2 \ln 4 - 8)$

(C) $2(c^2 \ln 5 - 8)$

(D) $c\ln 4 - 8$

$\textbf{Ans.} (B)$

$\textbf{Sol. }$ Let $\phi(x) = x^2\ln(4)-16\ln x,$ which is countinuous on $[4,5]$ and differentiable on $(4,5)$, so by LMVT,

$\dfrac{\phi(5)-\phi(4)}{5-4}=\phi'(c),$ $ c\in (4,5)$

Now, $\phi(5)-\phi(4)=\ln\left(\dfrac{4^{25}}{5^{16}}\right)$

and $\phi'(c)=\dfrac{2}{c}(c^2\ln 4 -8)$

$\implies c\ln\left(\dfrac{4^{25}}{5^{16}}\right) = 2(c^2\ln 4 - 8)$

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