Question

In [0, 1] Lagranges Mean Value Theorem is not applicable to

(a) f(x) = {((1/2 - x), " x < 1/2), ((1/2 - x)2, x ≥ 1/2)

(b) f(x) = {(sinx/x : x ≠ 0),(1, : x = 0)

(c) f(x) = x |x|

(d) f(x) = |x|

Answer 1

Correct option (a)

Explanation:

Clearly f(x) is not differentiable at x = 1/2

Thus, Lagranges Mean Value Theorem is not applicable.

Clearly f(x) is continuous in [0, 1] and differentiable in (0, 1

Thus, Lagranges Mean Value Theorem is applicable.

(c) We have f(x) = x|x| = x2 in [0, 1]

As we know that every polynomial function is continuous and differentiable everywhere.

So it is continuous in [0, 1] and differentiable in (0, 1).

Thus, Lagranges Mean Value Theorem is applicable.

(d) Also, f(x) = |x| = x

Clearly it is continuous in [0, 1] and differentiable in (0, 1).

Thus, Lagranges Mean Value Theorem is applicable.