Mean Value Theorem - Mathematics Questions and Answers

This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Mean Value Theorem”.

1. Function f should be _____ on [a,b] according to Rolle’s theorem.
a) continuous
b) non-continuous
c) integral
d) non-existent
View Answer

Answer: a
Explanation: According to Rolle’s theorem, if f : [a,b] → R is a function such that
i) f is continuous on [a,b]
ii) f is differentiable on (a,b)
iii) f(a) = f(b) then there exists at least one point c ∈ (a,b) such that f’(c) = 0

2. Function f is differential on (a,b) according to Rolle’s theorem.
a) True
b) False
View Answer

3. What is the relation between f(a) and f(b) according to Rolle’s theorem?
a) Equals to
b) Greater than
c) Less than
d) Unequal
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4. Does Rolle’s theorem applicable if f(a) is not equal to f(b)?
a) Yes
b) No
c) Under particular conditions
d) May be
View Answer

Answer: b
Explanation: According to Rolle’s theorem, if f : [a,b] → R is a function such that
i) f is continuous on [a,b]
ii) f is differentiable on (a,b)
iii) f(a) = f(b) then there exists at least one point c ∈ (a,b) such that f’(c) = 0

5. Another form of Rolle’s theorem for the differential condition is _____
a) f is differentiable on (a,ah)
b) f is differentiable on (a,a-h)
c) f is differentiable on (a,a/h)
d) f is differentiable on (a,a+h)
View Answer

Answer: d
Explanation: According to Rolle’s theorem, if f : [a,a+h] → R is a function such that
i) f is continuous on [a,a+h]
ii) f is differentiable on (a,a+h)
iii) f(a) = f(a+h) then there exists at least one θ c ∈ (0,1) such that f’(a+θh) = 0

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6. Another form of Rolle’s theorem for the continuous condition is _____
a) f is continuous on [a,a-h]
b) f is continuous on [a,h]
c) f is continuous on [a,a+h]
d) f is continuous on [a,ah]
View Answer

Answer: c
Explanation: According to Rolle’s theorem, if f : [a,a+h] → R is a function such that
i) f is continuous on [a,a+h]
ii) f is differentiable on (a,a+h)
iii) f(a) = f(a+h) then there exists at least one θ c ∈ (0,1) such that f’(a+θh) = 0

7. What is the relation between f(a) and f(h) according to another form of Rolle’s theorem?
a) f(a) < f(a+h)
b) f(a) = f(a+h)
c) f(a) = f(a-h)
d) f(a) > f(a+h)
View Answer

Answer: b
Explanation: According to Rolle’s theorem, if f : [a,a+h] → R is a function such that
i) f is continuous on [a,a+h]
ii) f is differentiable on (a,a+h)
iii) f(a) = f(a+h) then there exists at least one θ c ∈ (0,1) such that f’(a+θh) = 0

8. Function f is not continuous on [a,b] to satisfy Lagrange’s mean value theorem.
a) False
b) True
View Answer

Answer: a
Explanation: According to Lagrange’s mean value theorem, if f : [a,b] → R is a function such that
i) f is continuous on [a,b]
ii) f is differentiable on (a,b) then there exists a least point c ∈ (a,b) such that f’(c) = \(\frac {f(b)-f(a)}{b-a}\).

9. What are/is the conditions to satify Lagrange’s mean value theorem?
a) f is continuous on [a,b]
b) f is differentiable on (a,b)
c) f is differentiable and continuous on (a,b)
d) f is differentiable and non-continuous on (a,b)
View Answer

Answer: c
Explanation: According to Lagrange’s mean value theorem, if f : [a,b] → R is a function such that
i) f is continuous on [a,b]
ii) f is differentiable on (a,b) then there exists a least point c ∈ (a,b) such that f’(c) = \(\frac {f(b)-f(a)}{b-a}\).

10. Function f is differentiable on [a,b] to satisfy Lagrange’s mean value theorem.
a) True
b) False
View Answer

Answer: a
Explanation: According to Lagrange’s mean value theorem, if f : [a,b] → R is a function such that f is differentiable on (a,b) then there exists a least point c ∈ (a,b) such that f’(c) = \(\frac {f(b)-f(a)}{b-a}\). This shows Function f is differentiable on [a,b].

11. Lagrange’s mean value theorem is also called as _____
a) Euclid’s theorem
b) Rolle’s theorem
c) a special case of Rolle’s theorem
d) the mean value theorem
View Answer

Answer: d
Explanation: Lagrange’s mean value theorem is also called the mean value theorem and Rolle’s theorem is just a special case of Lagrange’s mean value theorem when f(a) = f(b).

12. Rolle’s theorem is a special case of _____
a) Euclid’s theorem
b) another form of Rolle’s theorem
c) Lagrange’s mean value theorem
d) Joule’s theorem
View Answer

Answer: c
Explanation: Rolle’s theorem is just a special case of Lagrange’s mean value theorem when f(a) = f(b) and Lagrange’s mean value theorem is also called the mean value theorem.

13. Is Rolle’s theorem applicable to f(x) = tan x on [ \(\frac {\pi }{4}, \frac {5\pi }{4}\) ]?
a) Yes
b) No
View Answer

Answer: b
Explanation: Given function is f(x) = tan x on [ \(\frac {\pi }{4}, \frac {5\pi }{4}\) ]
F(x) = tan x is not defined at x on [ \(\frac {\pi }{4}, \frac {5\pi }{4}\) ]
So, f(x) is not continuous on [ \(\frac {\pi }{4}, \frac {5\pi }{4}\) ].
Hence, Rolle’s theorem is not applicable.

14. What is the formula for Lagrange’s theorem?
a) f’(c) = \(\frac {f(a)+f(b)}{b-a}\)
b) f’(c) = \(\frac {f(b)-f(a)}{b-a}\)
c) f’(c) = \(\frac {f(a)+f(b)}{b+a}\)
d) f’(c) = \(\frac {f(a)-f(b)}{b+a}\)
View Answer

Answer: b
Explanation: According to Lagrange’s mean value theorem, if f : [a,b] → R is a function such that f is differentiable on (a,b) then the formula for Lagrange’s theorem is f’(c) = \(\frac {f(b)-f(a)}{b-a}\).

15. Find ’C’ using Lagrange’s mean value theorem, if f(x) = e^x, a = 0, b = 1.
a) e^e-1
b) e-1
c) log\(_e^{e+1}\)
d) log\(_e^{e-1}\)
View Answer