Given:

Now, we have to show that f(x) verify the Mean Value Theorem

First of all, Conditions of Mean Value theorem are:

a) f(x) is continuous at (a,b)

b) f(x) is derivable at (a,b)

If both the conditions are satisfied, then there exist some ‘c’ in (a,b) such that

Here,

On differentiating above with respect to x, we get

f'(x) = -1 × (4x – 1)^-1-1 × 4

⇒ f’(x) = -4 × (4x – 1)^-2

⇒f’(x) exist

Hence, f(x) is differentiable in (1,4)

We know that,

Differentiability ⇒ Continuity

⇒Hence, f(x) is continuous in (1,4)

Thus, Mean Value Theorem is applicable to the given function

Now,

x ∈ [1,4]

Now, let us show that there exist c ∈ (0,1) such that

On differentiating above with respect to x, we get

Put x = c in above equation, we get

…(i)

By Mean Value Theorem,

⇒

⇒ (4c – 1)² = 45

⇒ 4c – 1 = √45

⇒ 4c – 1 = ± 3√5

⇒ 4c = 1 ± 3√5

but

So, value of

Thus, Mean Value Theorem is verified.