Example 16
If a, b, are real numbers, show that there exists at
least one real number c
such that
Solution : Let f (x) = x^{3} over [a, b]
f (x) is continuous on [a, b] and differentiable in (a,
b). Also f (b) = b^{3} and f
' (x) = 3x^{2}.
Since f (x) satisfies all the conditions of meanvalue theorem, there
exists
at least one x=c, a < c < b such that
Example 17
Using mean  value theorem, prove that
Solution : Let f (x) = tan^{1}x
over [x_{2}, x_{1}]
Now f (x) is continuous on [x_{2}
, x_{1}] and differentiable in (x_{2},x_{1})
[next page]
