Mathematics Questions and Answers – Derivatives Application – Maxima and Minima

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Derivatives Application – Maxima and Minima”.

1. What is a monotonically increasing function?
a) x1 > x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b) ∀ c ∈ a
b) x1 < x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
c) x1 < x2 ⇒ f(x1) = f(x2) ∀ x1, x2 ∈ (a,b)
d) x1 = x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
View Answer

Answer: b
Explanation: A function f : (a,b) → R is said to be monotonically increasing on (a,b) if x1 < x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b). A monotonically increasing function can also be called as non-decreasing function.
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2. What is the mathematical expression for monotonically decreasing function?
a) x1 < x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
b) x1 < x2 ⇒ f(x1) ≥ f(x2) ∀ x1, x2 ∈ (a,b)
c) x1 = x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
d) x1 < x2 ⇒ f(x1) = f(x2) ∀ x1, x2 ∈ (a,b)
View Answer

Answer: b
Explanation: The definition of monotonically decreasing function is if a function f : (a,b) → R is said to be monotonically decreasing on (a,b) if x1 < x2 ⇒ f(x1) ≥ f(x2) ∀ x1, x2 ∈ (a,b). Hence, the mathematical expression is x1< x2 ⇒ f(x1) ≥ f(x2) ∀ x1, x2 ∈ (a,b).

3. What is the mathematical expression for a function to be strictly increasing on (a,b)?
a) x1 < x2 ⇒ f(x1) < f(x2) ∀ x1, x2 ∈ (a,b)
b) x1 < x2 ⇒ f(x1) ≥ f(x2) ∀ x1, x2 ∈ (a,b)
c) x1 = x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
d) x1 = x2 ⇒ f(x1) < f(x2) ∀ x1, x2 ∈ (a,b)
View Answer

Answer: a
Explanation: A function f : (a,b) → R is said to be strictly increasing on (a,b) if x1< x2 ⇒ f(x1) < f(x2) ∀ x1, x2 ∈ (a,b). (a,b) may be replaced by [a,b] or any interval in the definition.
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4. What is the mathematical expression for a function to be strictly decreasing on (a,b)?
a) x1 = x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
b) x1 < x2 ⇒ f(x1) > f(x2) ∀ x1, x2 ∈ (a,b)
c) x1 < x2 ⇒ f(x1) < f(x2) ∀ x1, x2 ∈ (a,b)
d) x1 < x2 ⇒ f(x1) = f(x2) ∀ x1, x2 ∈ (a,b)
View Answer

Answer: b
Explanation: There are two types of decreasing functions in maxima and minima. They are strictly decreasing and monotonically decreasing. The mathematical expression for strictly decreasing function is x1 < x2 ⇒ f(x1) > f(x2) ∀ x1, x2 ∈ (a,b).

5. A monotonic function on [a,b] is either a monotonically increasing or monotonically decreasing function.
a) False
b) True
View Answer

Answer: b
Explanation: Maximum and minimum values of a function are represented by monotonicity. A monotonic function on [a,b] is either a monotonically increasing or monotonically decreasing on [a,b].
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6. Monotonically increasing functions are usually referred to as decreasing functions.
a) True
b) False
View Answer

Answer: b
Explanation: Monotonically increasing functions are usually referred to as increasing functions whereas monotonically decreasing functions are usually referred to as decreasing functions.

7. What is the condition for a function f to be increasing if f be continuous and differentiable on (a,b)?
a) f’(x) < 0 ∀ x1, x2 ∈ (a,b)
b) f’(x) > 0 ∀ x1, x2 ∈ (a,b)
c) f’(x) = 0 ∀ x1, x2 ∈ (a,b)
d) f’(x) ≥ 0 ∀ x1, x2 ∈ (a,b)
View Answer

Answer: d
Explanation: If a function f be continuous and differentiable on (a,b), then f’(x) ≥ 0 ∀ x1, x2 ∈ (a,b) is the condition for the function f(x) to be increasing on (a,b).
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8. What is the condition for a function f to be decreasing if f be continuous and differentiable on (a,b)?
a) f’(x) > 0 ∀ x1, x2 ∈ (a,b)
b) f’(x) < 0 ∀ x1, x2 ∈ (a,b)
c) f’(x) = 0 ∀ x1, x2 ∈ (a,b)
d) f’(x) ≤ 0 ∀ x1, x2 ∈ (a,b)
View Answer

Answer: d
Explanation: If a function f be continuous and differentiable on (a,b), then f’(x) ≤ 0 ∀ x1, x2 ∈ (a,b) is the condition for the function f(x) to be decreasing on (a,b).

9. What is the condition for a function f to be strictly increasing if f be continuous and differentiable on (a,b)?
a) f’(x) > 0 ∀ x1, x2 ∈ (a,b)
b) f’(x) < 0 ∀ x1, x2 ∈ (a,b)
c) f’(x) ≤ 0 ∀ x1, x2 ∈ (a,b)
d) f’(x) = 0 ∀ x1, x2 ∈ (a,b)
View Answer

Answer: a
Explanation: The condition for a function ‘f’ to be strictly increasing is f’(x) > 0 ∀ x1, x2 ∈ (a,b) where the the function ‘f’ should be continuous and differentiable on (a,b).
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10. What is the condition for a function f to be strictly decreasing if f be continuous and differentiable on (a,b)?
a) f’(x) > 0 ∀ x1, x2 ∈ (a,b)
b) f’(x) < 0 ∀ x1, x2 ∈ (a,b)
c) f’(x) = 0 ∀ x1, x2 ∈ (a,b)
d) f’(x) ≤ 0 ∀ x1, x2 ∈ (a,b)
View Answer

Answer: b
Explanation: The mathematical expression for strictly decreasing function is f’(x) < 0 ∀ x1, x2 ∈ (a,b). This is the condition for strictly decreasing function and only possible when function ‘f’ is continuous and differentiable on (a,b).

11. What is the condition for a function f to be constant if f be continuous and differentiable on (a,b)?
a) f’(x) > 0 ∀ x1, x2 ∈ (a,b)
b) f’(x) < 0 ∀ x1, x2 ∈ (a,b)
c) f’(x) = 0 ∀ x1, x2 ∈ (a,b)
d) f’(x) ≤ 0 ∀ x1, x2 ∈ (a,b)
View Answer

Answer: c
Explanation: One of the properties of a function is to be constant. A function is said to be constant when it satisfies the condition f’(x) = 0 ∀ x1, x2 ∈ (a,b) where the function ‘f’ should be continuous and differentiable on (a,b).

12. What is the mathematical expression of non-decreasing function?
a) x1 > x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b) ∀ c ∈ a
b) x1 < x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
c) x1 < x2 ⇒ f(x1) = f(x2) ∀ x1, x2 ∈ (a,b)
d) x1 = x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
View Answer

Answer: b
Explanation: A function f : (a,b) → R is said to be monotonically increasing on (a,b) if x1 < x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b). A monotonically increasing function can also be called as non-decreasing function.

13. What is the mathematical expression for monotonically non-increasing function?
a) x1 < x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
b) x1 < x2 ⇒ f(x1) ≥ f(x2) ∀ x1, x2 ∈ (a,b)
c) x1 = x2 ⇒ f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a,b)
d) x1 < x2 ⇒ f(x1) = f(x2) ∀ x1, x2 ∈ (a,b)
View Answer

Answer: b
Explanation: The meaning of a monotonic function is it either never decreases or never increases. The condition for a function to be monotonically non-increasing is x1 < x2 ⇒ f(x1) ≥ f(x2) ∀ x1, x2 ∈ (a,b).

14. What is the relation between f(x) and ℓ when the maximum value or greatest value function f is defined on a set A and ℓ ∈ f(A)?
a) f(x) < ℓ ∀ x ∈ A
b) f(x) ≤ ℓ ∀ x ∈ A
c) f(x) = ℓ ∀ x ∈ A
d) f(x) > ℓ ∀ x ∈ A
View Answer

Answer: b
Explanation: A function f defined on a set A and ℓ ∈ f(A), then ℓ is the maximum or the greatest value of f in A if f(x) ≤ ℓ ∀ x ∈ A and the minimum or the least value of f in A if f(x) ≥ ℓ ∀ x ∈ A.

15. What is the relation between f(x) and ℓ when the minimum value or least value function f is defined on a set A and ℓ ∈ f(A)?
a) f(x) < ℓ ∀ x ∈ A
b) f(x) ≤ ℓ ∀ x ∈ A
c) f(x) ≥ ℓ ∀ x ∈ A
d) f(x) > ℓ ∀ x ∈ A
View Answer

Answer: c
Explanation: The relation between f(x) and ℓ when the minimum value or least value function f is f(x) ≥ (ℓ) ∀ x ∈ A where the function is defined on a set A and ℓ ∈ f(A).

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter