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📝 Optimization on open and infinite intervals (25 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 25 questions available

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What is Optimization on open and infinite intervals?

Definition:
On open or infinite intervals, absolute extrema are not guaranteed. Find critical points and analyze limits at boundaries. If the function grows without bound, no absolute extremum exists; if it approaches a finite limit, compare with critical values to determine if an extremum is attained.

Example:
Minimize $f(x) = x + \frac{1}{x}$ for $x > 0$ . $f'(x) = 1 - \frac{1}{x^2} = 0 \Rightarrow x=1$ . $f(1)=2$ , and limits at $0, \infty$ are $\infty$ , so min is $2$ .

Reason:
Requires careful limit analysis to ensure the critical point is indeed a global extremum and not just a local one in unbounded domains.

Easy

Medium

Hard

📝 All Optimization on open and infinite intervals MCQs

Q1. Which of the following describes a closed interval?

A.[a,b] ✅

B.(a,b)

C.(-\infty,a]

D.[a,\infty)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A closed interval contains both of its endpoints, so the notation $[a,b]$ includes a and b. The other choices either omit endpoints or extend to infinity, which are not closed in the usual real‑number sense.

Q2. What is the notation for the interval extending from a finite number a to positive infinity, including a?

A.(a,\infty)

B.(-\infty,a]

C.[a,\infty)

D.(a,\infty] ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The interval that starts at a finite point a and continues without bound to the right, while including a, is written $[a,\infty)$ . This notation indicates the left endpoint is closed and the right side is unbounded.

Q3. If a function f is continuous on the closed interval [0,5] and differentiable on (0,5), which statement must be true?

A.f(0)=f(5)

B.There exists c in (0,5) with f'(c)=0

C.f is increasing on [0,5] ✅

D.f'(0)=f'(5)

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: By the Mean Value Theorem, a function continuous on a closed interval and differentiable on its interior must have at least one interior point c where the instantaneous rate of change equals the average rate, i.e., f'(c)=0 when the endpoint values are equal.

Q4. Consider the function g defined on (-∞,2]. If lim_{x→2⁻} g(x)=7 and g(2)=7, which of the following is guaranteed?

A.g is continuous at 2 ✅

B.g has a maximum at 2

C.g is differentiable at 2

D.g is bounded on (-∞,2]

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Continuity at a point requires that the limit from the left equals the function value at that point. Since both are 7, g satisfies the definition of continuity at x=2, regardless of its behavior elsewhere.

Q5. Let h be continuous on [-∞,3] (interpreted as (-∞,3]). If lim_{x→-∞} h(x)=4 and h(3)=10, which Intermediate Value property must hold?

A.There exists c with h(c)=7 ✅

B.h attains a global minimum

C.h is monotonic

D.No conclusion can be drawn

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The Intermediate Value Theorem applies to continuous functions on intervals that may be infinite. Since h takes the values 4 (as x→-∞) and 10 at x=3, every value between 4 and 10, such as 7, must be assumed at some point in the interval.

Q6. Compare the closure of the interval (1,4) with the interval [1,4]. Which statement is correct?

A.Both sets are equal

B.Closure of (1,4) is (1,4)

C.Closure of (1,4) is [1,4] ✅

D.Closure of (1,4) is (1,4]

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The closure of a set adds all its limit points. The open interval (1,4) lacks its endpoints, which are limit points, so its closure is the closed interval [1,4].

Q7. Which of the following intervals is both closed and unbounded?

A.[0,∞)

B.(-∞,5] ✅

C.[-∞,∞]

D.None of the above

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An interval is closed if it contains its endpoint(s). The set (-∞,5] includes its right endpoint 5 while extending without bound to the left, making it closed and unbounded.

Q8. Given the function p(x)=1/x defined on (-∞,0)∪(0,∞), which of the following statements about its intervals of continuity is true?

A.Continuous on each of the two intervals ✅

B.Continuous on the whole real line

C.Discontinuous only at x=0

D.Not continuous on any interval

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The function 1/x is continuous everywhere it is defined. Its domain consists of two separate intervals, and on each of those intervals the standard limit definition of continuity is satisfied.

Q9. Apply the Extreme Value Theorem to f(x)=√x on the interval [0,9]. What can be concluded?

A.f attains both a maximum and a minimum ✅

B.Only a maximum

C.Only a minimum

D.No extrema exist

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The Extreme Value Theorem guarantees that a continuous function on a closed, bounded interval achieves both its greatest and least values. Since √x is continuous on [0,9], it has a minimum at x=0 (value 0) and a maximum at x=9 (value 3).

Q10. Explain why the interval (-∞,∞) is considered closed in the topology of the extended real line.

A.It contains its endpoints

B.Its complement is empty

C.It is both open and closed

D.All of the above ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: In the extended real line, the whole space is always both open and closed. Its complement is the empty set, which is trivially open, and the set itself contains the added points -∞ and ∞, satisfying the definition of a closed set.

Q11. Synthesize the relationship between the domain being a closed interval and the guarantee of a function's global extrema, using the function q(x)=x³-3x on [-2,2].

A.The global max occurs at x=1

B.The global min occurs at x=-1

C.Both global extrema occur at interior critical points ✅

D.Both global extrema occur at the endpoints only

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The function q has critical points at x=±1 where q'(x)=0. Evaluating q at those points and at the endpoints shows the largest and smallest values are attained at both interior points and endpoints, illustrating that closed intervals allow extrema to appear anywhere within the domain.

Q12. If a function is defined on the interval [a,∞) and its derivative is always positive, which of the following must be true?

A.The function is increasing ✅

B.The function is decreasing

C.The function is constant

D.The function has a maximum

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A positive derivative indicates a locally increasing behavior at every point of the interval. Since the interval extends to infinity, the function cannot turn downward, so it must be monotonically increasing throughout.

Q13. Which interval is the complement of the closed interval [3,7] in ℝ?

A.(-∞,3)∪(7,∞) ✅

B.[-∞,3)∪(7,∞]

C.(-∞,3]∪[7,∞)

D.[3,7]

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Removing the closed interval [3,7] from the real line leaves all numbers less than 3 and greater than 7. Both of those pieces are open, giving the union (-∞,3)∪(7,∞).

Q14. When applying the Mean Value Theorem on the interval (-∞,4], which condition fails?

A.Continuity on the interval

B.Differentiability on the interior

C.Existence of finite endpoints ✅

D.None of the above

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The Mean Value Theorem requires a finite closed interval [a,b] with both endpoints real numbers. The interval (-∞,4] lacks a left endpoint, so the hypothesis involving a closed, bounded interval is not satisfied.

Q15. Suppose f is continuous on [-∞,0] and lim_{x→-∞} f(x)=5. If f(0)=5, what can be inferred about the existence of a point c where f(c)=5?

A.Such c must exist ✅

B.No such c exists

C.c must be 0

D.Infinite many such c

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: By the Intermediate Value Theorem, a continuous function that approaches 5 as x→-∞ and actually equals 5 at x=0 must attain the value 5 somewhere in the interval; the theorem guarantees at least one such point.

Q16. Compare the set of limit points of the interval (0,1) with its closure.

A.Same as the interval

B.Closure adds 0 and 1 ✅

C.Closure adds only 0

D.Closure adds only 1

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The limit points of (0,1) are all points that can be approached by sequences inside the interval, which include every point of (0,1) plus the endpoints 0 and 1. Adding those endpoints yields the closure [0,1].

Q17. Apply Rolle's Theorem to f(x)=sin x on the interval [0,π]. Which conclusion is correct?

A.There exists c with f'(c)=0 ✅

B.No such c exists

C.f is not continuous

D.f is not differentiable

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Rolle's Theorem applies because sin x is continuous on [0,π] and differentiable on (0,π), with equal endpoint values sin 0 = sin π = 0. Hence there exists at least one c in (0,π) where the derivative cos c = 0, i.e., c = π/2.

Q18. Given g defined on [-∞,5] with g continuous and g(5)=0, and lim_{x→-∞} g(x)=10. Using the Intermediate Value Theorem, which value must g take at some point in the interval?

A.5

B.0

C.10 ✅

D.7

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Since g continuously moves from the value 10 as x→-∞ to the value 0 at x=5, every intermediate value between 10 and 0 must be assumed. The number 5 lies between them, so there exists some c with g(c)=5.

Q19. Which of the following statements about the set S = {x ∈ ℝ : x ≤ 3} is true regarding its topological properties?

A.S is open but not closed

B.S is closed but not bounded ✅

C.S is bounded and open

D.S is neither open nor closed

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The set consists of all real numbers less than or equal to 3. Its complement (3,∞) is open, making S closed. Because it extends indefinitely to the left, S is unbounded.

Q20. Synthesize how the concept of a closed interval [a,b] ensures the existence of a supremum for any bounded above subset of the interval.

A.By completeness of real numbers ✅

B.By Archimedean property

C.By density of rationals

D.By Bolzano–Weierstrass theorem

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The real numbers are complete: every non‑empty set that is bounded above has a least upper bound (supremum). A closed interval contains its endpoints, so any subset bounded above by b will have its supremum within the interval, guaranteeing existence.

Q21. What is the term for an interval that includes neither of its endpoints?

A.Closed interval

B.Open interval ✅

C.Half‑open interval

D.Unbounded interval

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An interval that excludes both endpoints is called an open interval, denoted by parentheses, e.g., $(a,b)$ . This contrasts with closed intervals that include endpoints and half‑open intervals that include exactly one endpoint.

Q22. If a function h is defined on (-∞,0] and satisfies h'(x)=2 for all x in (-∞,0), what can be said about h on that interval?

A.h is constant

B.h is linear with slope 2 ✅

C.h is quadratic

D.h is undefined

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Integrating the constant derivative h'(x)=2 yields h(x)=2x+C, a linear function with slope 2. This relationship holds for every point in the interval where the derivative is defined.

Q23. Compare the behavior of the function f(x)=1/x on the intervals (-∞,0) and (0,∞). Which statement is accurate?

A.Both intervals yield increasing functions

B.Both decreasing ✅

C.First decreasing, second increasing

D.First increasing, second decreasing

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The derivative f'(x) = -1/x² is negative for all non‑zero x, indicating that f is strictly decreasing on both the left and right sides of the vertical asymptote at 0.

Q24. Apply the definition of a limit at infinity to determine lim_{x→∞} (2x+3)/(x-4).

A.0

B.1

C.2 ✅

D.Does not exist

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Dividing numerator and denominator by x gives (2+3/x)/(1-4/x). As x→∞, the terms 3/x and 4/x vanish, leaving 2/1 = 2? Wait correction: the ratio of leading coefficients is 2/1 = 2, but the answer choice B is 1. Actually the correct limit is 2. Therefore the correct answer should be C. Adjusted:

Q25. Given the function k(x)=√(x²+1) defined on (-∞,∞), which of the following statements about its monotonicity on the interval (-∞,0] is correct?

A.k is increasing

B.k is decreasing ✅

C.k is constant

D.k is non‑decreasing but not strictly

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The derivative k'(x)=x/√(x²+1) is negative for all x<0 and equals 0 at x=0. Hence k decreases strictly on (-∞,0) and levels off at 0, making the function decreasing on the entire interval (-∞,0].

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