Discuss the applicability of Rolle’s theorem for the following function on the indicated interval:

f(x) = |x| on [–1, 1]

We have,    f (x) = |x| =  

Since a polynomial function is everywhere continuous and differentiable. Therefore, f (x) is continuous and differentiable for all x < 0 and for all x > 0 except possibly at x = 0.

So, consider the point x = 0.

We have,

 

 

Thus, f (x) is continuous at x = 0

Hence, f (x) is continuous on [–1, 1]

Now,

(LHD at x = 0) =  

Þ (LHD at x = 0) =          [Q f (x) = – x for x < 0 and f (0) = 0]

and, (RHD at x = 0) =  [Q f (x) = x for x ³ 0]

 

\ (LHD at x = 0) ¹ RHD at x = 0

This shows that f (x) is not differentiable at x = 0 Î (– 1, 1). Thus, the condition of derivability at each point of (– 1, 1) is not satisfied.

Hence, Rolle’s theorem is not applicable to f (x) = |x| on [–1, 1]