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

Condition 1:

Firstly, we have to show that f(x) is continuous.

Here, f(x) is continuous because f(x) has a unique value for each x ∈ [1,5]

Condition 2:

Now, we have to show that f(x) is differentiable

[using chain rule]

∴ f’(x) exists for all x ∈ (1,5)

So, f(x) is differentiable on (1,5)

Hence, Condition 2 is satisfied.

Thus, mean value theorem is applicable to given function.

Now,

Now, we will find f(a) and f(b)

so, f(a) = f(1)

and f(b) = f(5)

Now, let us show that c ∈ (1,5) such that

On differentiating above with respect to x, we get

Put x = c in above equation, we get

By Mean Value theorem,

Squaring both sides, we get

⇒ 16c² = 24 × (25 – c²)

⇒ 16c² = 600 – 24c²

⇒ 24c² + 16c² = 600

⇒ 40c² = 600

⇒ c² = 15

⇒ c = √15 ∈ (1,5)

Hence, Mean Value Theorem is verified.