use lagrange multipliers to maximize & minimize the funct subject to the constraint. F(x,y)=4x+3y; x2+y2=25?

Answer 1

see http://www.math.uri.edu/~bkaskosz/trans/lagrange1.html

Answer 2

In

4x + 3y + λ ( x² + y² - 25 )

set partial derivatives to 0:

x: 4 + λ ( 2x ) = 0

y: 3 + λ ( 2y ) = 0

Substitute x and y into x² + y² - 25 and solve to get

λ² = 1/4, or λ = ± ½

If λ = ½ then x = -4, y = -3 and F(-4,-3) = -25 < 0, so is an absolute minimum.

If λ = -½ then x = 4, y = 3 and F(4,3) = 25 > 0, so is an absolute maximum.