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

Explanation

(i) `f(x)=[x], " " x in [5,9]`
f(x) = [x] is discontinuous for integral values.
` :. ` Lagrange's theorem is not applicable here.
(ii) `f(x)=[x], " " x in [-2,2]`
f(x) = [x] is discontinuous for integral values.
` :. ` Lagrang's theorem is not applicable here.
(iii) `f(x)=1-x^(2), " " x in [1,2]`
`f(x)=1-x^(2)` is polynomial function.
` :. f(x)` is continuous in [1, 2] and differentiable in (1, 2)
Thus, f(x) satisfies all conditions of Langrange's theorem in [1, 2].
Now, `f'(x)= -2x`
Let `f'(c)=(f(2)-f(1))/(2-1)`
`implies-2c=[1-(2)^(2)]-[1-(1)^(2)]`
`implies-2c= -3`
`impliesc= (3)/(2) in (1, 2)`
Hence, Lagrange's theorem is verified.