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

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

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

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

View Answer

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Explanation: Given f(x) = e

^{x}, a = 0, b = 1

f’(c) = \(\frac {f(b)-f(a)}{b-a}\)

e

^{c}= \(\frac {e-1}{1-0}\)

e

^{c}= e – 1

C = log\(_e^{e-1}\)

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