# Q6 Examine the applicability of Mean Value Theorem for all three functions given in        the above exercise 2.

According to Mean value theorem function
must be
a )  continuous in given closed interval say [a,b]
b ) differentiable in given open interval say (a,b)
Then their exist a   such that

If all these conditions are satisfies then we can verify  mean value theorem
Given function is

It is clear that Given function  is not  continuous for each and every point in [5,9]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,

Now,
R.H.L. at x = n ,

We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (5,9)
Hence, Mean value theorem is not applicable for given function       ,

Similaly,
Given function is

It is clear that Given function  is not  continuous for each and every point in [-2,2]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,

Now,
R.H.L. at x = n ,

We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (-2,2)
Hence, Mean value  theorem is not applicable for given function       ,

Similarly,
Given function is

Now, being a polynomial , function  is continuous in [1,2] and differentiable in(1,2)
Now,

And

Now,

Now,

And
Therefore, mean value theorem is applicable for the function

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