Answer :

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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