Minimum Sum with a Fixed Product Optimization Calculator

It returns each optimal number and the minimum sum, using the exact equality condition behind the arithmetic-geometric mean inequality. The result is deterministic and immediate, making it useful for calculus optimization exercises, algebra checks, geometric design problems, and automated answer generation without relying on a numerical search or an arbitrary starting guess.

Set up the fixed-product optimization problem

Suppose two positive numbers are called x and y, and their fixed product is P. The constraint is xy = P, while the quantity to minimize is the sum S = x + y. Enter P in the product field; it must be a finite number strictly greater than zero because the problem specifically concerns two positive real numbers. The constraint lets you replace y with P/x, turning the sum into the one-variable function S(x) = x + P/x for x greater than zero. This form makes the tradeoff visible. Choosing x very small forces y to become very large, while choosing x very large makes x itself dominate the sum. Somewhere between those extremes lies a unique balance point. The calculator reports that point as first_number and second_number, followed by minimum_sum. The labels distinguish the requested quantities even though the two optimal values are equal. Units are not imposed by the service; if the product represents square meters, squared centimeters, or another compound unit, carry the corresponding base unit into each number and the sum. Only the mathematical scalar values are returned.

Why equal numbers produce the global minimum

There are two standard ways to justify the answer. With calculus, differentiate S(x) = x + P/x to obtain S'(x) = 1 - P/x squared. Setting the derivative equal to zero gives x squared = P, and positivity selects x = square root of P. Substitution into y = P/x gives the same positive square root for y. The second derivative, 2P/x cubed, is positive throughout the allowed domain, so this critical point is a strict minimum. The behavior near zero and toward infinity confirms that it is the global minimum rather than merely a local one. The arithmetic-geometric mean inequality provides an even shorter proof: for positive x and y, (x + y)/2 is at least the square root of xy. Because xy equals P, every allowed sum is at least 2 square root of P, and equality occurs exactly when x equals y. The calculator uses this closed-form equality condition directly. It does not sample candidates or iterate toward an approximation, so there is no search interval, convergence tolerance, or possibility of selecting the wrong local extremum.

Read the result and apply it correctly

For a product of 36, the optimal values are 6 and 6, and the minimum sum is 12. An unequal valid pair such as 4 and 9 still has product 36, but its sum is 13, which illustrates the penalty for moving away from equality. Products that are not perfect squares work the same way: both optimal values may be irrational and are represented with standard JavaScript floating-point arithmetic. The service deliberately does not round them, because a classroom interface may want four decimal places while a downstream calculation may need the full available precision. Round only when presenting the answer, and avoid feeding a prematurely rounded value back into a strict product check. The result applies over positive real numbers. If a problem restricts the choices to integers, imposes lower or upper bounds, uses negative numbers, or asks about three or more factors, its feasible set is different and this calculator is not the right solver. In geometry, the formula often appears when a rectangle has fixed area and perimeter must be minimized: equal side lengths produce the square. In calculus exercises, use the response to check your result, but still show the constraint, substitution, derivative or inequality, equality condition, and domain reasoning required by the written solution.