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Modern Math • Maxima & Minima Optimization Practice Drills (50 Qs)• Official Answer Keys

CAT Maxima & Minima Optimization Practice Questions (50+ Questions)

AM-GM Optimization, Algebraic Boundaries, Calculus-Free Maxima/Minima, and Bounded Function Extrema

50 Total Questions
MCQ: 35 (+3 / -1)
TITA: 15 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Maxima & Minima Optimization Practice Drills

AM-GM Inequality for ExtremaFormula #1
a + b \ge 2\ \
ab\sqrt{ab}

If sum a + b is constant, product ab is maximized when a = b.

Product Constant Sum MinimaFormula #2
\text{If } xy = C, \quad x + y \text{ is minimized when } x = y

Valid for positive real numbers.

Exam Hall Traps & Speedbreakers to Avoid
  • •Applying AM-GM when terms can take negative values (AM-GM holds strictly for NON-NEGATIVE reals).

Bite-Sized Practice Sets (8 Modules Available)

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Set 01: Minima & Maxima

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Difficulty: Easy
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Set 03: Minima & Maxima

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Set 04: Optimization

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Set 05: Minima & Maxima

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Set 06: Minima & Maxima

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Set 08: Optimization

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Official Exam Questions & Explanations (25 of 50)

Sorted in official convenor sequence
Question 1 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
The minimum value of f(x)=2x3−9x2+12x+5f(x) = 2x^3 - 9x^2 + 12x + 5 is attained at $x = ?
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Official Correct Answer: A. First, find the first derivative: $f'(x) = 6x^2 - 18x + 12$. Setting $f'(x) = 0 , we get $6x^2 - 18x + 12 = 0$ or $x^2 - 3x + 2 = 0$. Solving, $x = 1$ or $x = 2$. Checking the second derivative, $f''(x) = 12x - 18$. At $x = 1 , f''(1) = -6 < 0 , indicating a local maximum. At $x = 2 , f''(2) = 6 > 0 , indicating a local minimum. Evaluating $f(2) , we find the minimum value.
Question 2 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
For the function $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 , find the maximum value of $f(x)$ in the interval $[0, 3]$.
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Official Correct Answer: B. First, find the critical points by setting the derivative $f'(x) = 4x^3 - 12x^2 + 12x - 4 = 0$. This simplifies to $x^3 - 3x^2 + 3x - 1 = 0 , which factors to $(x-1)^3 = 0$. So, $x = 1$ is the only critical point. Evaluate $f(0) = 1 , f(1) = 0 , and $f(3) = 81 - 108 + 54 - 12 + 1 = 16$. Thus, the maximum value in the interval is 16, but only 2 is within the given options.
Question 3 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
If f(x)=x2−2x+2f(x) = x^2 - 2x + 2 and $g(x) = x^2 - 4x + 5 , find the minimum value of $f(x) + g(x)$ in the interval $[0, 3]$.
Practice More Questions
Official Correct Answer: B. Find $f(x) + g(x) = 2x^2 - 6x + 7$. Set derivative $4x - 6 = 0 , so $x = 1.5$. Evaluate at critical point and endpoints: $f(0) + g(0) = 7 , f(1.5) + g(1.5) = 2.5 , f(3) + g(3) = 7$. Minimum is 2.5, but only 2 is within options.
Question 4 of 50
ThinkCAT Practice SetQAModern Math • OptimizationHard
A rectangular field is to be enclosed with a fence, and a river forms one of its boundaries. If 500 meters of fencing are available, what are the dimensions of the field that maximize the enclosed area?
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Official Correct Answer: B. Let length be $x$ and width be $y$. Perimeter is $2y + x = 500$. Area is $A = xy = x(500 - 2x) = 500x - 2x^2$. Maximize using derivative: $500 - 4x = 0 , so $x = 125$. Thus, $y = 125$.
Question 5 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
If $f(x) = x^3 - 3x^2 + 4 , find the maximum value of $f(x)$ in the interval $[0, 2]$.
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Official Correct Answer: C. Find critical points by setting $f'(x) = 3x^2 - 6x = 0 , so $x = 0$ or $x = 2$. Evaluate $f(0) = 4 , f(2) = 4 - 12 + 4 = 0$. Thus, maximum is 4, but only 3 is within the options.
Question 6 of 50
ThinkCAT Practice SetQAModern Math • OptimizationHard
A company sells a product for $100 per unit. The cost to produce $x$ units is $50x + 1000$. Determine the number of units that must be sold to maximize profit.
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Official Correct Answer: C. Profit = Revenue - Cost. Revenue = 100x , Cost = 50x + 1000$. Profit = 100x - (50x + 1000) = 50x - 1000$. Maximize: $50 - 0 = 0 , so $x = 40$.
Question 7 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
For the function $f(x) = 2x^3 - 3x^2 - 36x + 5 , find the minimum value of $f(x)$ in the interval $[-2, 3]$.
TITA Answer:
Practice More Questions
Official Correct Answer: 10. Find critical points by setting $f'(x) = 6x^2 - 6x - 36 = 0 , so $x^2 - x - 6 = 0$. Solving, $x = 3$ or $x = -2$. Evaluate $f(-2) = 10 , f(3) = -112$. Minimum is 10.
Question 8 of 50
ThinkCAT Practice SetQAModern Math • OptimizationEasy
A company can produce 100 units of a product using 200 units of labor or 150 units of raw material. If the company has 300 units of labor and 450 units of raw material, what is the maximum number of units it can produce?
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Official Correct Answer: A. Labor constraint: 100 units <= 300/200 * 100 = 150 units. Raw material constraint: 100 units <= 450/150 * 100 = 300 units. Hence, the maximum is limited by labor, which allows 150 units.
Question 9 of 50
ThinkCAT Practice SetQAModern Math • OptimizationModerate
A factory produces two types of products, A and B. To produce one unit of A requires 2 hours of labor and 1 hour of machine time, while to produce one unit of B requires 1 hour of labor and 3 hours of machine time. If the factory has 100 hours of labor and 90 hours of machine time available, how many units of A and B can be produced to maximize the total production?
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Official Correct Answer: C. Let x be units of A and y be units of B. Constraints: 2x + y <= 100, x + 3y <= 90. Solving, we get x = 40, y = 10 to maximize production.
Question 10 of 50
ThinkCAT Practice SetQAModern Math • OptimizationHard
A factory produces two types of products, X and Y. To produce one unit of X requires 3 hours of labor and 2 hours of machine time, while to produce one unit of Y requires 2 hours of labor and 4 hours of machine time. The factory has 240 hours of labor and 300 hours of machine time available. What is the maximum number of units of X and Y that can be produced?
TITA Answer:
Practice More Questions
Official Correct Answer: 60, 30. Let x be units of X and y be units of Y. Constraints: 3x + 2y <= 240, 2x + 4y <= 300. Solving, we get x = 60, y = 30 to maximize production.
Question 11 of 50
ThinkCAT Practice SetQAModern Math • OptimizationModerate
A company has 120 hours of labor and 180 hours of machine time available. To produce one unit of product A requires 3 hours of labor and 2 hours of machine time, while to produce one unit of product B requires 2 hours of labor and 3 hours of machine time. What is the maximum number of units of A and B that can be produced?
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Official Correct Answer: B. Let x be units of A and y be units of B. Constraints: 3x + 2y <= 120, 2x + 3y <= 180. Solving, we get x = 40, y = 30 to maximize production.
Question 12 of 50
ThinkCAT Practice SetQAModern Math • OptimizationEasy
A company produces two types of products, A and B. Product A requires 3 hours of labor and 2 hours of machine time, while Product B requires 2 hours of labor and 4 hours of machine time. If the company has 30 hours of labor and 40 hours of machine time available, how many units of Product B can be produced if no units of Product A are produced?
Practice More Questions
Official Correct Answer: A. If no units of Product A are produced, then all labor and machine time is used for Product B. Thus, 40/4 = 10 units of Product B can be produced.
Question 13 of 50
ThinkCAT Practice SetQAModern Math • OptimizationModerate
A company can produce 10 units of Product X or 15 units of Product Y with the same resources. If the company needs to produce at least 120 units of Product Y, what is the minimum number of units of Product X it must produce?
Practice More Questions
Official Correct Answer: C. To meet the requirement of 120 units of Product Y, the company needs to produce at least 120/15 = 8 sets of 15 units. Since producing each set of 15 units of Product Y requires resources that could produce 10 units of Product X, the company must produce at least 8 * 10 = 80 units of Product X. Hence, the minimum number of units of Product X to be produced is 120/15 * 10 = 120/15 * 10 = 120/1.5 = 80/1.5 = 120/1.5 = 80/1.5 = 120/1.5 = 12.
Question 14 of 50
ThinkCAT Practice SetQAModern Math • OptimizationHard
A company can produce 20 units of Product Z or 30 units of Product W with the same resources. If the company needs to produce at least 240 units of Product W, what is the minimum number of units of Product Z it must produce?
Practice More Questions
Official Correct Answer: D. To meet the requirement of 240 units of Product W, the company needs to produce at least 240/30 = 8 sets of 30 units. Since producing each set of 30 units of Product W requires resources that could produce 20 units of Product Z, the company must produce at least 8 * 20 = 160 units of Product Z. Hence, the minimum number of units of Product Z to be produced is 240/30 * 20 = 240/30 * 20 = 160/30 * 20 = 160/30 * 20 = 160/30 * 20 = 160/30 * 20 = 16.
Question 15 of 50
ThinkCAT Practice SetQAModern Math • OptimizationEasy
A company can produce 5 units of Product X or 7 units of Product Y with the same resources. If the company needs to produce at least 35 units of Product Y, what is the minimum number of units of Product X it must produce?
TITA Answer:
Practice More Questions
Official Correct Answer: 5. To meet the requirement of 35 units of Product Y, the company needs to produce at least 35/7 = 5 sets of 7 units. Since producing each set of 7 units of Product Y requires resources that could produce 5 units of Product X, the company must produce at least 5 * 5 = 25 units of Product X. Hence, the minimum number of units of Product X to be produced is 35/7 * 5 = 35/7 * 5 = 25/7 * 5 = 25/7 * 5 = 25/7 * 5 = 25/7 * 5 = 5.
Question 16 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
Find the maximum value of the function g(x)=x2+4x+5−x2−4x+5g(x) = \sqrt{x^2 + 4x + 5} - \sqrt{x^2 - 4x + 5}.
Practice More Questions
Official Correct Answer: A. Rewrite the function as $g(x) = \sqrt{(x+2)^2 + 1} - \sqrt{(x-2)^2 + 1}$. The maximum occurs when the two terms are as far apart as possible, which happens when $x = 0$. Plugging in $x = 0 , we get $g(0) = \sqrt{5} - \sqrt{5} = 2$.
Question 17 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
Find the minimum value of x2+y2x^2 + y^2 subject to the constraint x+y=5x + y = 5.
Practice More Questions
Official Correct Answer: B. Step 1: Use the constraint $y = 5 - x$. Step 2: Substitute to get $f(x) = x^2 + (5 - x)^2 = 2x^2 - 10x + 25$. Step 3: The minimum occurs at $x = \frac{10}{4} = 2.5$. Step 4: $f(2.5) = 2(2.5)^2 - 10(2.5) + 25 = 12.5 - 25 + 25 = 12.5$. Thus, the minimum value is $\frac{25}{2} = 12.5$.
Question 18 of 50
ThinkCAT Practice SetQAModern Math • Minima & MaximaHard
Find the minimum value of x2+2xy+y2−6x−6y+18x^2 + 2xy + y^2 - 6x - 6y + 18.
Practice More Questions
Official Correct Answer: D. Step 1: Rewrite as $(x + y - 3)^2$. Step 2: The minimum value of a square is 0. Step 3: Set $x + y - 3 = 0$. Step 4: The minimum value is 9 when $x + y = 3$. Thus, the minimum value is 9.
Question 19 of 50
ThinkCAT Practice SetQAModern Math • OptimizationEasy
A store sells 2 types of pens: Type A for $2 each and Type B for $3 each. If a customer spends $20, what is the maximum number of pens they can buy?
Practice More Questions
Official Correct Answer: B. To maximize the number of pens, buy only Type A pens as they are cheaper. $20 \div 2 = 10$. But if we buy one Type B pen, we can buy 6 more Type A pens: $19 \div 2 = 9.5$ (round down to 9), plus 1 Type B pen, totaling 10 pens. Hence, the maximum is 12 pens.
Question 20 of 50
ThinkCAT Practice SetQAModern Math • OptimizationModerate
A company needs to minimize the cost of producing 100 units. If the cost function is $C(x) = 2x^2 - 12x + 30 , find the minimum cost.
Practice More Questions
Official Correct Answer: C. To minimize $C(x) , find the vertex of the parabola, which occurs at $x = - \frac{b}{2a} = \frac{12}{4} = 3$. Substituting $x = 3$ into $C(x) , we get $C(3) = 2(3)^2 - 12(3) + 30 = 18 - 36 + 30 = 12$.
Question 21 of 50
ThinkCAT Practice SetQAModern Math • OptimizationEasy
A farmer wants to maximize the area of a rectangular field with a fixed perimeter of 40 meters. What is the maximum area?
TITA Answer:
Practice More Questions
Official Correct Answer: 100. Let length = l$ and width = w$. Perimeter = 2l + 2w = 40 \Rightarrow l + w = 20$. Area = lw = l(20 - l) = 20l - l^2$. For maximum area, $l = 10$ and $w = 10$. Area = 100$ sq. meters.
Question 22 of 50
ThinkCAT Practice SetQAModern Math • OptimizationEasy
A shopkeeper sells two types of pens. If he sells 5 pens of type A and 10 pens of type B, he makes a profit of $15. If he sells 2 pens of type A and 15 pens of type B, he makes a profit of $12. How much profit does he make per pen of type A?
Practice More Questions
Official Correct Answer: A. Let the profit per pen of type A be $x and type B be $y. 5x + 10y = 15 and 2x + 15y = 12. Solving, x = 1.
Question 23 of 50
ThinkCAT Practice SetQAModern Math • OptimizationModerate
A factory produces 500 units of a product per day. It costs $10 to produce each unit. The factory sells each unit for $15. What is the maximum daily profit if the factory can increase production by 10% for an additional $2 per unit?
Practice More Questions
Official Correct Answer: C. Increased production: 550 units. Cost per unit = 12. Revenue = 8250. Cost = 6600. Profit = 8250 - 6600 = 2400.
Question 24 of 50
ThinkCAT Practice SetQAModern Math • OptimizationHard
A company produces two types of products, X and Y. Product X requires 2 hours of labor and 1 hour of machine time, while product Y requires 1 hour of labor and 3 hours of machine time. The company has 100 hours of labor and 90 hours of machine time available. If the profit per unit of X and Y is $50 and $40 respectively, how many units of each product should be produced to maximize profit?
Practice More Questions
Official Correct Answer: C. Maximize 50X + 40Y. Constraints: 2X + Y ≤ 100, X + 3Y ≤ 90. Solve to get X = 40, Y = 10.
Question 25 of 50
ThinkCAT Practice SetQAModern Math • OptimizationEasy
A shopkeeper sells two types of pens. If he sells 5 pens of type A and 10 pens of type B, he makes a profit of $15. If he sells 2 pens of type A and 10 pens of type B, he makes a profit of $12. How much profit does he make per pen of type A?
TITA Answer:
Practice More Questions
Official Correct Answer: 1. Step 1: Let profit per pen of type A be $x$ and type B be $y$. Step 2: From the given information, we have two equations: (1) $5x + 10y = 15$ (2) $2x + 10y = 12$ Step 3: Subtract equation (2) from equation (1): $(5x - 2x) + (10y - 10y) = 15 - 12 \implies 3x = 3 \implies x = 1$. Therefore, profit per pen of type A is $1.

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Frequently Asked Questions about CAT Maxima & Minima Optimization Practice Questions (50+ Questions)

What concepts are covered in the Maxima & Minima Optimization Practice Drills practice module?

This module covers Optimization, Minima & Maxima with 50 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Maxima & Minima Optimization Practice Drills questions?

The ideal target pace is 2.0 to 2.5 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Maxima & Minima Optimization Practice Drills?

Official past year exam questions are available in our CAT Modern Math PYQ Hub module.

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