Here you will learn mean value theorems i.e rolle’s theorem, lagrange’s theorem and extreme value theorem.
Let’s begin –
Mean Value Theorems
(a) Rolle’s Theorem
Let f be a real valued function defined on the closed interval [a, b] such that
(i) it is continuous on the closed interval [a, b],
(ii) it is differentiable on the open interval (a, b)
(iii) f(a) = f(b)
Then, there exist a real number c \(\in\) (a, b) such that f'(c) = 0
Note : If f is differentiable function then between any two consecutive roots of f(x) = 0, there is atleast one root of the equation f'(x) = 0
(b) Lagrange’s Mean Value Theorem (LMVT)
Let f be a function that satisfies the following conditions :
(i) f is continuous in [a, b]
(ii) f is differentiable in (a, b).
Then there is a number c in (a, b) such that f'(c) = \(f(b) – f(a)\over b – a\)
Extreme Value Theorem
If f is continuous on [a, b] then f takes on, a least value m and a greatest value M on this interval.
Note : Continuity throught the interval [a, b] is essential for the validity of this theorem.
(a) If a continuous function y = f(x) is increasing in the closed interval [a, b] , then f(a) is the least value and f(b) is the greatest value of f(x) in [a, b]
(b) If a continuous function y = f(x) is decreasing in the closed interval [a, b] , then f(b) is the least value and f(a) is the greatest value of f(x) in [a, b]
(c) If a continuous function y = f(x) is increasing/decreasing in the (a, b) , then no greatest and least value exist.
