Question

Let f(x) = ln x and g(x) = x². If c Î (4, 5) then c ln $\left( \frac{4^{25}}{5^{16}} \right)$ equals to –

• A. c ln 5 – 8
• B. 2 (c² ln 4 – 8)
• C. 2 (c² ln 5 – 8)
• D. ln 4 – 8)

Answer 1

Answer: (B)

2 (c² ln 4 – 8)

Answer 2

f

Let f(x) = x² ln (4) – 16 ln x, which is continuous on [4, 5] and differentiable on (4, 5), so by LMVT, $\frac{\varphi(5) - \varphi(4)}{5 - 4}$

= f¢(3), c Î (4, 5)

Now, f(5) – f(4) = ln $\left( \frac{4^{25}}{5^{16}} \right)$ and f¢(3) = $\frac{2}{c}$ (c² ln 4 – 8)

̃ c ln $\left( \frac{4^{25}}{5^{16}} \right)$ = 2 (c² ln 4 – 8).