EXAMPLE 5 Find the maximum value of the function f(x, y, z) = x + 2y + 9z on the curve of intersection of the plane x − y + z = 1 and the cylinder x2 + y2 = 1. SOLUTION We maximize the function f(x, y, z) = x + 2y + 9z subject to the constraints g(x, y, z) = x − y + z = 1 and h(x, y, z) = x2 + y2 = 1. The Lagrange condition is ∇f = λ∇g + μ∇h, so we solve the equations

The Lagrangian,

has critical points where its partial derivatives vanish:

tells us , so that

Then with , we get

and tells us

Then there are two critical points, . The critical point with the negative -coordinates gives the maximum value, .

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step-by-step explanation:

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