Optimization with Two Variables

Business Calculus · Axiom Academy

LESSON Optimization of Functions of Two Variables Finding maxima and minima in multivariable settings In business, we often want to find the best combination of two inputs that maximizes profit or minimizes cost. With functions of two variables z = f(x, y), we look for points where the surface reaches a peak (maximum) or valley (minimum). At a critical point, both partial derivatives equal zero simultaneously Set both f x = 0 and f y = 0, then solve the system of equations Find f xx , f yy , and f xy at each critical point Compute D = f xx · f yy - (f xy )² Apply the Second Derivative Test Use D and f xx to classify the critical point Surface curves downward in all directions Surface curves upward in all directions Surface curves up in one direction, down in another Left: Maximum (inverted bowl) | Center: Minimum (bowl) | Right: Saddle Find and classify all critical points of Since D = 8 > 0 and f xx = 4 > 0, the point (2, 3) is a local minimum . A company's profit depends on the prices of two products: D = 15 > 0 and f xx = -6 Local Maximum The optimal prices are p₁ = 30 and p₂ = 25, yielding maximum profit. The negative second partials (-6 and -4) indicate diminishing returns to price increases, which is typical in competitive markets. When D = 0: The test is inconclusive; further analysis needed Boundary values: For constrained problems, also check boundaries Global vs. Local: A local extremum may not be the global optimum

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