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

1. What is a monotonically increasing function?

a) x_{1} > x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b) ∀ c ∈ a

b) x_{1} < x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

c) x_{1} < x_{2} ⇒ f(x_{1}) = f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

d) x_{1} = x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

Explanation: A function f : (a,b) → R is said to be monotonically increasing on (a,b) if x

_{1}< x

_{2}⇒ f(x

_{1}) ≤ f(x

_{2}) ∀ x

_{1}, x

_{2}∈ (a,b). A monotonically increasing function can also be called as non-decreasing function.

2. What is the mathematical expression for monotonically decreasing function?

a) x_{1} < x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

b) x_{1} < x_{2} ⇒ f(x_{1}) ≥ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

c) x_{1} = x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

d) x_{1} < x_{2} ⇒ f(x_{1}) = f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

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 x

_{1}< x

_{2}⇒ f(x

_{1}) ≥ f(x

_{2}) ∀ x

_{1}, x

_{2}∈ (a,b). Hence, the mathematical expression is x

_{1}< x

_{2}⇒ f(x

_{1}) ≥ f(x

_{2}) ∀ x

_{1}, x

_{2}∈ (a,b).

3. What is the mathematical expression for a function to be strictly increasing on (a,b)?

a) x_{1} < x_{2} ⇒ f(x_{1}) < f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

b) x_{1} < x_{2} ⇒ f(x_{1}) ≥ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

c) x_{1} = x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

d) x_{1} = x_{2} ⇒ f(x_{1}) < f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

Explanation: A function f : (a,b) → R is said to be strictly increasing on (a,b) if x

_{1}< x

_{2}⇒ f(x

_{1}) < f(x

_{2}) ∀ x

_{1}, x

_{2}∈ (a,b). (a,b) may be replaced by [a,b] or any interval in the definition.

4. What is the mathematical expression for a function to be strictly decreasing on (a,b)?

a) x_{1} = x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

b) x_{1} < x_{2} ⇒ f(x_{1}) > f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

c) x_{1} < x_{2} ⇒ f(x_{1}) < f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

d) x_{1} < x_{2} ⇒ f(x_{1}) = f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

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 x

_{1}< x

_{2}⇒ f(x

_{1}) > f(x

_{2}) ∀ x

_{1}, x

_{2}∈ (a,b).

5. A monotonic function on [a,b] is either a monotonically increasing or monotonically decreasing function.

a) False

b) True

View Answer

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].

6. Monotonically increasing functions are usually referred to as decreasing functions.

a) True

b) False

View Answer

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 ∀ x_{1}, x_{2} ∈ (a,b)

b) f’(x) > 0 ∀ x_{1}, x_{2} ∈ (a,b)

c) f’(x) = 0 ∀ x_{1}, x_{2} ∈ (a,b)

d) f’(x) ≥ 0 ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

Explanation: If a function f be continuous and differentiable on (a,b), then f’(x) ≥ 0 ∀ x

_{1}, x

_{2}∈ (a,b) is the condition for the function f(x) to be increasing on (a,b).

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 ∀ x_{1}, x_{2} ∈ (a,b)

b) f’(x) < 0 ∀ x_{1}, x_{2} ∈ (a,b)

c) f’(x) = 0 ∀ x_{1}, x_{2} ∈ (a,b)

d) f’(x) ≤ 0 ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

Explanation: If a function f be continuous and differentiable on (a,b), then f’(x) ≤ 0 ∀ x

_{1}, x

_{2}∈ (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 ∀ x_{1}, x_{2} ∈ (a,b)

b) f’(x) < 0 ∀ x_{1}, x_{2} ∈ (a,b)

c) f’(x) ≤ 0 ∀ x_{1}, x_{2} ∈ (a,b)

d) f’(x) = 0 ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

Explanation: The condition for a function ‘f’ to be strictly increasing is f’(x) > 0 ∀ x

_{1}, x

_{2}∈ (a,b) where the the function ‘f’ should be continuous and differentiable on (a,b).

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 ∀ x_{1}, x_{2} ∈ (a,b)

b) f’(x) < 0 ∀ x_{1}, x_{2} ∈ (a,b)

c) f’(x) = 0 ∀ x_{1}, x_{2} ∈ (a,b)

d) f’(x) ≤ 0 ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

Explanation: The mathematical expression for strictly decreasing function is f’(x) < 0 ∀ x

_{1}, x

_{2}∈ (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 ∀ x_{1}, x_{2} ∈ (a,b)

b) f’(x) < 0 ∀ x_{1}, x_{2} ∈ (a,b)

c) f’(x) = 0 ∀ x_{1}, x_{2} ∈ (a,b)

d) f’(x) ≤ 0 ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

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 ∀ x

_{1}, x

_{2}∈ (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) x_{1} > x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b) ∀ c ∈ a

b) x_{1} < x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

c) x_{1} < x_{2} ⇒ f(x_{1}) = f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

d) x_{1} = x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

Explanation: A function f : (a,b) → R is said to be monotonically increasing on (a,b) if x

_{1}< x

_{2}⇒ f(x

_{1}) ≤ f(x

_{2}) ∀ x

_{1}, x

_{2}∈ (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) x_{1} < x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

b) x_{1} < x_{2} ⇒ f(x_{1}) ≥ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

c) x_{1} = x_{2} ⇒ f(x_{1}) ≤ f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

d) x_{1} < x_{2} ⇒ f(x_{1}) = f(x_{2}) ∀ x_{1}, x_{2} ∈ (a,b)

View Answer

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 x

_{1}< x

_{2}⇒ f(x

_{1}) ≥ f(x

_{2}) ∀ x

_{1}, x

_{2}∈ (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

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

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).

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