Question

For $m>1, n>1$, the value of $c$ for which the Rolle's theorem is applicable for the function $f(x)=x^{2 m-1}(a-x)^{2 n}$ in $(0, a)$ is

• A. $\frac{2am - 1}{ m + 2n - 1}$
• B. $\frac{a(m - n + 1)}{ 2m + 2n}$
• C. $\frac{a( 2m - 1)}{ 2m + 2n - 1 }$
• D. $\frac{a( 2m + 1)}{m + n - 1 }$

Answer 1

Answer: (C)

Since it is given that Rolle's theorem is applicaple for the function
$f(x)=x^{2 m-1}(a-x)^{2 n}$ in $(0, a) .$
So, $f'(x)=(2 m-1) x^{2 m-2}(a-x)^{2 n}-2 n(a-x)^{2 n-1} x^{2 m-1}$
at $x=c, m>1, n>1$
$f'(c)=0$
$\Rightarrow (2 m-1) c^{2 m-2}=2 n c^{2 m-1}(a-c)^{2 n-1}$
$\Rightarrow \frac{(2 m-1)}{c}=\frac{2 n}{a-c}$
$\Rightarrow c=\frac{a(2 m-1)}{2 n+2 m-1}$