Q. 765.0( 2 Votes )

Verify mean value theorem for each of the functions given


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



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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