Mean Value Theorems
Rolle’s Theorem
If a function f (x) is such that:
- f (x) is continuous in the closed interval a ≤ x ≤ b and
- f ′(x) exists for every point in the open interval a < x < b and
- f(a) = f(b),
Then there exists at least one value of x, say c, where a < c < b such that f′(c) = 0.
Note: Rolle’s theorem will not hold good.
- If f(x) is discontinuous at some point in the interval a < x < b
- If f′(x) does not exist at some point in the interval a < x < b or
- If f(a) ≠ f(b)
Lagrange’s Mean Value Theorem
If a function f(x) is:
- Continuous in closed interval a ≤ x ≤ b and
- Differentiable in open interval (a, b) i.e., a < x < b,
then there exist at least one value c of x lying in the open interval a < x < b such thatf′(c) = f(b) – f(a) / b – a
Cauchy’s Mean Value Theorem
If f(x) and g(x) are two real-valued functions defined on [a, b] and,
- f(x) and g(x) are continuous in [a, b]
- f(x) and g(x) are derivable in (a, b) and
- g′(x) ≠ 0 ∀ x ∈ (a, b)
Then there exists least one point c ∈ (a, b), such that,f'(c) / g'(c) = f(b) – f(a) / g(b) – g(a)
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