# Verify mean value theorem for each of the functions given 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.

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