Maxima and Minima in CAT Quant: AM-GM and Symmetry

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Questions that ask for the largest area, the smallest sum or the greatest product look like calculus problems. In CAT, they almost never need calculus. They need one inequality and a sense of symmetry, and the answer arrives in a couple of lines.

This guide covers the two ideas that solve most of them: AM-GM, and a symmetry rule that says what happens when a sum or a product is held fixed. It also covers the trap that catches careful students, which is applying the bound where it cannot actually be reached.

What Maxima and Minima Questions Ask

The family is wide in wording and narrow in method. A farmer fences a field and wants the largest area. A product of two numbers with a fixed sum is to be maximised. The minimum value of an expression such as x plus 9 over x is asked for. In each case there is a quantity to optimise and a constraint that ties the variables together.

In CAT, the constraint is usually simple, and the expression is built so that one clean inequality finishes the job. That is why recognising the shape matters more than any heavy technique. Once you see that the expression is a sum of two positive terms whose product is constant, the minimum is already known.

You can also solve many such problems by trying a few values, but it is slow and unreliable. A method that gives the extreme value with a reason is better, because it also tells you where the extreme occurs, and some questions ask for exactly that.

AM-GM in One Paragraph

For two positive numbers a and b, the arithmetic mean (a plus b) divided by 2 is at least the geometric mean, the square root of ab. Equality holds only when a equals b. For three positive numbers, the same holds with the cube root of abc, and so on for more terms.

The two statements you will use most are these. First, a plus b is at least 2 times the square root of ab, so a sum has a floor once the product is known. Second, ab is at most the square of (a plus b) divided by 2, so a product has a ceiling once the sum is known.

Both statements are the same inequality read in two directions. If you remember one, you can rebuild the other in a few seconds. The condition that matters is positivity, since for negative numbers the inequality can fail.

The Symmetry Rule

The symmetry rule is a plain-language version of AM-GM that is easy to remember under pressure. If the sum of two positive quantities is fixed, their product is largest when they are equal. If their product is fixed, their sum is smallest when they are equal.

Take a rectangle with perimeter 40, so the two sides add to 20. The area is the product of the sides, and the sum is fixed, so the area is largest when both sides are 10. The largest area is 100. Now reverse it: a rectangle of area 64 has a perimeter that is smallest when it is a square of side 8, so the least perimeter is 32.

The same logic extends to three quantities. If a plus b plus c equals 12, then abc is at most 64, reached at 4, 4 and 4. In words, balance is best: spreading a fixed total evenly gives the biggest product, and splitting a fixed product evenly gives the smallest total.

What is fixed	What you optimise	Best when parts are
Sum of two positives	Largest product	Equal
Product of two positives	Smallest sum	Equal
Sum of three positives	Largest product	Equal
Perimeter of a rectangle	Largest area	A square

Worked Examples

Example one. Find the minimum value of x plus 9 over x for positive x. The two terms multiply to 9, which is constant. So the sum is at least 2 times the square root of 9, which is 6, and equality needs x equal to 9 over x, so x is 3. The minimum is 6, at x equal to 3.

Example two. The numbers x and y are positive with 2x plus 3y equal to 12. Find the largest value of xy. Treat 2x and 3y as the two parts. Their sum is 12, so their product 6xy is at most 36, and xy is at most 6. Equality needs 2x equal to 3y, each equal to 6, so x is 3 and y is 2. Check: 2 times 3 plus 3 times 2 is 12, and xy is 6.

Example three. For x greater than 1, find the minimum of x plus 4 over (x-1). Write t for x-1, so the expression is t plus 4 over t plus 1. For positive t, t plus 4 over t is at least 4, so the whole expression is at least 5, reached at t equal to 2, so x is 3. The trick was to shift the variable so the two terms became a pair with a constant product.

Notice that each example needed a small rewrite before AM-GM applied. The skill is not the inequality. It is shaping the expression into two positive terms with either a fixed sum or a fixed product.

The Trap: Is Equality Reachable?

AM-GM gives a bound. It does not guarantee that the bound is the actual minimum or maximum. The bound is reached only when the equal-parts condition can be met inside the allowed range of the variable.

Take x plus 1 over x, where x lies between 2 and 5. AM-GM says the sum is at least 2, with equality at x equal to 1. But x cannot be 1 in this range. The expression is increasing for x above 1, so its minimum in the range is at x equal to 2, where it is 2.5. The bound of 2 was true but not reachable.

The check takes seconds. Solve the equality condition, and see whether the value lies in the allowed range. If it does, the bound is the answer. If it does not, look at the endpoints of the range, since the extreme will sit there. Examiners set this trap on purpose, and many answer options are built around the unreachable bound.

When Another Method Is Faster

AM-GM is not the only tool. For a quadratic, the vertex gives the extreme directly. For the expression -x squared plus 8x plus 3, the maximum occurs at x equal to 4, and the value is -16 plus 32 plus 3, which is 19. There is no need for an inequality.

For modulus expressions, think of distances on a number line. The expression |x-2| plus |x-6| is the sum of distances from x to 2 and to 6. It is smallest when x lies between 2 and 6, and the minimum value is 4, the gap between them. A sketch of the number line is faster than any algebra.

When the expression is a ratio or a product of several terms that do not pair up, a graph or a check of a few values can be the quickest route, and in the exam an estimate that eliminates options may be enough. For help choosing between algebra and trial, see our guide on algebraic versus numerical methods in CAT Quant.

Connecting to Inequalities and Functions

Maxima and minima questions share ground with two other chapters. Inequalities supply the tools, such as the sign of an expression and the ranges where it is positive. Functions supply the picture, since the maximum or minimum is the highest or lowest point of a graph.

If your inequalities are shaky, the extreme value questions will feel harder than they are. Our guide to inequalities and the wavy curve method covers the sign work, and the article on quadratic equations formulas and PYQs covers the vertex and the discriminant.

Once the tools are in place, practise functions questions to build the habit of reading a graph quickly. A short session each day on this mix tends to work better than a long one, because the patterns need repeated exposure to stick.

A Practice Routine

Start with ten minutes of single idea questions: five that use AM-GM and five that use the symmetry rule. Say aloud which quantity is fixed before you write anything. Then add a second set where the equality condition is not reachable, so you practise the check.

In the next session, mix quadratics and modulus questions in with AM-GM ones, and decide for each one which method is quickest. Track your times. If a question takes more than about three minutes without a clear route, park it and return only if time remains. Sequence by return on effort.

A CAT study planner can hold these short sessions, and you can talk to a CAT mentor if you want someone to look at the questions you keep missing. The formulas are short. The skill is spotting the shape.