Question

In [0, 1] Lagrange's mean value theorem is NOT applicable to
[IIT Screening 2003]

• A. $f\left(x\right)\{\begin{array}{cc}\frac{1}{2}-x,x<& \frac{1}{2}\\ {(\frac{1}{2}-x)}^2,x\ge \frac{1}{2}& \end{array}$
• B. $f\left(x\right)\{\begin{array}{cc}\frac{sinx}{x},x\ne 0& \\ 1,x=0& \end{array}$
• C. f(x) = x |x|
• D. f(x) = |x|

Answer 1

The correct option is A $f\left(x\right)\{\begin{array}{cc}\frac{1}{2}-x,x<& \frac{1}{2}\\ {(\frac{1}{2}-x)}^2,x\ge \frac{1}{2}& \end{array}$

$f\left(x\right)\{\begin{array}{cc}\frac{1}{2}-x,x<& \frac{1}{2}\\ {(\frac{1}{2}-x)}^2,x\ge \frac{1}{2}& \end{array}$
On Verification you'll find that the function is not differentiable at x = 1/2.
RHD = 0 whereas LHD = -1 at x = 1/2