question_answer

Verify Lagrange's mean value theorem for \[f(x)=\log \,\,x\] in [1, 2].

Answer:

We have, \[f(x)=\log \,\,x,\] where \[x\in [1,\,\,2]\] As we know that log x is continuous in its domain \[(0,\,\,\infty ),\] therefore f(x) is continuous in [1, 2]         Also, \[f'(x)=\frac{1}{4},\] which exist for all \[x\in (1,\,\,2)\] Thus, \[f(x)\] is continuous in [1, 2] and differentiable Now, consider \[f'(c)=\frac{f(b)-f(a)}{b-a}=\frac{f(2)-f(1)}{2-1}\]             \[=\frac{\log 2-\log 1}{1}=\log 2\]           \[[\because \,\,log\,\,1=0]\] \[\Rightarrow \]   \[\frac{1}{c}=\log 2\] \[\Rightarrow \] \[c=\frac{1}{\log 2}={{\log }_{2}}e\in (1,\,\,2)\] Hence, Lagrange's mean value theorem is verified.