Q. 735.0( 1 Vote )

# 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 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.

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