A value of c for which the conclusion of Mean Value Theorem holds for the function f(x) = logex on the interval [1, 3] is

Question

A value of c for which the conclusion of Mean Value Theorem holds for the function f(x) = logex on the interval [1, 3] is (A) 2 log3e (B) (1/2)loge3 (C) log3e (D) loge3

Tardigrade Admin · Accepted Answer

Using mean value theorem f '(c)=(f (3)-f (1)/3-1) ⇒ (1/c)=(log 3-log 1/2) ⇒ c=(2/loge 3)=2loge e.

Q. A value of $c$ for which the conclusion of Mean Value Theorem holds for the function $f (x) = l o g_{e} x$ on the interval $[1, 3]$ is

$2 l o g_{3} e$

$\frac{1}{2} l o g_{e} 3$

$l o g_{3} e$

$l o g_{e} 3$

Using mean value theorem
$f^{'} (c) = \frac{f ( 3 ) - f ( 1 )}{3 - 1}$
$\Rightarrow \frac{1}{c} = \frac{l o g 3 - l o g 1}{2}$
$\Rightarrow c = \frac{2}{l o g _{e} 3} = 2 l o g_{e} e .$

Q. A value of c for which the conclusion of Mean Value Theorem holds for the function f(x)=loge​x on the interval [1,3] is

2log3​e

21​loge​3

log3​e

loge​3