# Mathematics Questions and Answers – Mean Value Theorem

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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

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

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

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
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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

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)

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]

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)

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

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)

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

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

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

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

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) = $$\frac {f(a)-f(b)}{b+a}$$

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) = ex, a = 0, b = 1.
a) ee-1
b) e-1
c) log$$_e^{e+1}$$
d) log$$_e^{e-1}$$

Explanation: Given f(x) = ex, a = 0, b = 1
f’(c) = $$\frac {f(b)-f(a)}{b-a}$$
ec = $$\frac {e-1}{1-0}$$
ec = e – 1
C = log$$_e^{e-1}$$

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