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Rolle’s theorem

If a function f(x) is
  1. continuous in the closed interval [a, b], i.e., continuous at each point in the interval [a, b]
  2. differentiable in an open interval (a, b), i.e., differentiable at each point in the open interval (a, b)
  3. f(a) = f(b)
then there will be at least one point c, in the interval (a, b) such that f′(c) = 0.
 
91196.png

 

Note: Converse of Rolle’s theorem is not true, i.e., if a function f(x) is such that f′(c) = 0 for at least one c in the open interval (a,b) then it is not necessary that
  1. f(x) is continuous in [ab]
  2. f(x) is differentiable in (ab)
  3. f(a) = f(b)
For example, we consider the function f(x) = x3 – x2 – x + 1 and the interval [–1, 2].
 
Here, f′(x) = 3x2 – 2x – 1 ∴ f′(1) = 3 – 2 – 1 = 0 and I ∈ (–1, 2)
 
But condition (3) of Rolle’s theorem is not satisfied since f(–1) ≠ f(2).
 

Lagrange’s mean value theorem

If a function f(x) is
  1. continuous in the closed interval [a, b], i.e., continuous at each point in the interval [a, b]
  2. differentiable in the open interval (a, b), i.e., differentiable at each point in the interval (a, b)
then there will be at least one point c, where a < c < b such that
f′(c) = 91165.png
 
91159.png




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