Page 51 - Mathematics Class - XII

Viva-Voce

1. Define Lagrange’s Mean Value Theorem.
Ans. If a function f (x) is (i) continuous in the closed interval [a, b] and (ii) differentiable in the open

fb() − fa()
interval (a, b), then there exists some point c ∈ [a, b] such that f ′ (c) = .
ba−
2. What is the use of mean value theorem in real life?
Ans. Speeding tickets are a wonderful application of the mean value theorem in traffic law.

3. What is/are the difference between Lagrange’s Mean Value Theorem and Rolle’s theorem?
Ans. Mean Value Theorem Rolle’s Theorem

I. Definition Function f is continuous on a closed interval Function f is continuous on a closed
[a, b] and differentiable on the open interval interval [a, b] and differentiable on
(a, b). Then there is at least one point c in the open interval (a, b). If f (a) = f (b).
(a, b) where fb () fa() Then there is at least one point c in
fc() (a, b) where f ′(c) = 0

ba
II. Tangent At point c, the tangent is parallel to the secant At point c, the tangent is parallel to
joining (a, f (a)) and (b, f (b)). the x-axis.

III. Difference It is the first mean value theorem. It is also It is a special case of mean value
known as Lagrange’s Mean Value Theorem. theorem.

4. What is the geometrical meaning of Mean Value Theorem?
Ans. Geometrically, Mean Value Theorem states that there exists at least one point c in (a, b) such that the
tangent at the point (c, f (c)) is parallel to the chord joining the points (a, f (a)) and (b, f (b)).

MCQs

1. A value of c for which the mean value theorem holds for the function f (x) = log x in the interval [1, 3] is
e
a) 2 log e b) 1/2 log 3 c) log e d) log 3
3 e 3 e
2. The value of c in mean value theorem for the function f (x) = x (x – 2), x ∈ [1, 2] is

a) 3 b) 2 c) 1 d) 5
2 3 2 2
3. The value of c in mean value theorem for the function f (x) = (x – 3) (x – 6) (x – 9), in [3, 5] is

a) 6 13 b) 6 13 c) 6 13 d) None of these

3 3 3
Answers: 1. a) 2 log e 2. a) 3 3. c) 6 13
3
2 3

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