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, .

x = ± 6i

step-by-step explanation:

x^2 = -36

take the square root of each side

sqrt(x^2) = sqrt(-36)

we know sqrt(ab) = sqrt(a) sqrt(b)

x = ±sqrt(36) sqrt(-1)

we know that sqrt (-1) = i

x = ± 6i

a. she will receive a refund of $278.

step-by-step explanation: