Skip to content
We use only essential cookies to keep you signed in and remember your preferences. See our Cookie Policy.
TestStop
MocksPracticeResourcesIndiaBlogPricing
Sign in
Questions › Mathematics › Single-variable optimisation

Single-variable optimisation: previous year questions

Asked 37 times in 6 exams across 14 years. Newest first.

Practise Single-variable optimisation, easy to hard →

Jump to a subtopic
  • Unconstrained extrema and the first- and second-order conditions 12
  • Optimisation on a closed interval 13
  • Applied optimisation 12

Unconstrained extrema and the first- and second-order conditions

Finding and classifying maxima and minima using derivatives.

  1. IIT JAM 2023 Q59For k∈ℝ, let f(x)=x⁴+2x³+kx²-k. If x=3/2 is a local minimum of f and m is the global minimum value, then f(0)-m is (in integer).
  2. CUET-PG 2020 Q64The function y = f(x) reaches a (relative) minimum where
  3. JNU 2020 Q43f(x) = x⁴ - 4x³ + 4x² + 4 Find a local minimum and a local maximum of the function f(x).
  4. UoH 2019 Q50What is the minimum value of the function f(x) = 4x² - 4x + 1?
  5. CUET-PG 2018 Q45Let y=x⁴-6x³+4x²-13. At x=4, the function is
  6. UoH 2018 Q36The function f(x) = x² – 12x + 36x will attain a ……… at when x = 6
  7. JNU 2017 Q35Let f(x) = x³ - 3x² + 3. On what interval is the function decreasing?
  8. DSE 2016 Q16Consider f:ℝ→ℝ defined by f(x)=Σᵢ₌₁¹¹|x-aᵢ| for all x∈ℝ, where a₁<…<a₁₁. This function has a minimum when x equals:
  9. JNU 2015 Q49f(x) = (2x - 3)³
  10. DSE 2014 Q27Suppose c is a given positive real number. The equation ln x=cx² must have a solution if
  11. DSE 2012 Q46What is the total number of local maxima and local minima of the following function f(x)={ (2+x)³ if -3<x≤ -1; x^(2/3) if -1<x<2 }
  12. DSE 2011 Q1For the real-valued function f(x)=x⁴-4x³+6x²-4x+1 (defined for all real numbers x), the point x=1 is

Optimisation on a closed interval

Global maxima and minima when the domain is bounded, including endpoints and existence.

  1. UGC-NET 2023 Q24Suppose f(.) is a continuous function over a closed and bounded interval [a, b]. Then there exists a point d in [a, b] where f(.) has a minimum, and…
  2. UGC-NET 2023 Q65Consider the following three Constrained Optimization Problems (COP): (a) Maximize z = ln x subject to x >= 1; (b) Maximize z = ln x subject to x <=…
  3. JNU 2021 Q20Let f(x) = x², and the domain of x is [1, 2]. Which of the following statements is true?
  4. DSE 2019 Q3The maximum value attained by the function f(x)=x³-x²-x-1 on the set S={x| x²-x-2≤ 0} occurs at
  5. JNU 2018 Q24Let y = f(x), where x ∈ [a, b] and b > a, be a continuously differentiable function. Suppose f(x) attains maximum at x = x*. Then it is always true…
  6. JNU 2017 Q48Find the global maximum of f(x) = x³ - 3x in the interval [-2, 3].
  7. JNU 2016 Q1Determine the absolute minimum for the following function and interval: g(x) = 2x³ + 3x² - 12x + 4 on [-4, 2]
  8. JNU 2015 Q46If you maximize f(x) = 2/x³, subject to 0 < x < 1, then the maximum value of f(x) is obtained at
  9. DSE 2013 Q23Consider f(x)=9/2x^(2/3)-3/5x^(5/3) for all x in [-1,5]. f(5) is approximately 4.386. f attains a maximum on [-1,5] at:
  10. JNU 2012 Q63Let f be a twice differentiable real-valued function defined on the set of all real numbers greater than or equal to 0 and less than or equal to 1.…
  11. JNU 2012 Q96Suppose f(x) = a₁ x + a₂ + (a₃ / x), where a₁, a₂ and a₃ are positive constants and x is assumed to take only positive values. Which of the…
  12. DSE 2010 Q38Consider a twice differentiable function f:ℜ→ℜ and a,b∈ℜ with a<b, f(a)=0=f(b) and D²f(x)+Df(x)-1=0 for every x∈[a,b]. Then,
  13. DSE 2010 Q39Consider f as described in the previous question. Then,

Applied optimisation

Maximising profit, revenue or area and minimising cost or time in applied settings.

  1. UGC-NET 2025 Q78A company has determined that total cost C, total revenue R and the number of units produced X are related as C = 100 + 0.015 X^2 and R = 3X. Find…
  2. UoH 2021 Q76If the demand function for the product of a firm is p = √(1350 - 6q), find the output level that maximizes total revenue.
  3. UoH 2020 Q41If the total revenue and total cost function is given by R(q) = 10q - 0.01q² and C(q) = 100 + 5q. Find the level of output that will maximize profit
  4. JNU 2016 Q4We want to construct a box with a square base and we only have 6 m² of material to use in construction of the box. Assume that all the materials are…
  5. JNU 2016 Q5Determine the point(s) on y = x² + 1 that is/are closest to (0, 2).
  6. DSE 2014 Q28Sania's boat is at point A; the closest point on land, B, is 2 km away, and C on land is 6 km from B with triangle ABC right-angled at B. She rows…
  7. DSE 2013 Q26A six meter long string is cut into two pieces. The first piece, of length x, is used to make a circle; the second, of length 6-x, to make a square.…
  8. UoH 2013 Q90At what level of output q, the average cost (AC) will be minimum when the AC function is given by AC = q² - 5q + 8? (Hint: Apply mathematical method…
  9. DSE 2012 Q48A rectangle has its lower left hand corner at the origin and its upper right hand corner on the graph of f(x)=x²+(1/x²). For which x is the area of…
  10. UoH 2012 Q38The level of output q, at which the average cost (AC) will be minimum, where the AC = q² - 5q + 8 is,
  11. DSE 2011 Q23The closest point on the parabola y=1/4x² from a given point (0,b) on the vertical axis, with b>0, is the origin if and only if
  12. JNU 2010 Q57Suppose a plant can be used to produce in a day x units of product 1 and y units of product 2 where y = (32 - 5x)/(10 - x), where 32/5 ≥ x ≥ 0. If…