find the minimum value of x+y, subject to the constraints xy=16, x>0, y>0
Solution
Let f(x,y) = x + y and g(x, y) = xy. ?f = <1, 1> and ?g = . The system of equations
to solve is ?f = ??g, g = 16. The first two equations give x = y = 1/?. Plugging this into the
constraint gives 1/?² = 16 ==> ? = ±1/4 ==> x = y = ±4. There are two optimal values, one max
and one min. The max is 8 when (x,y) = (4, 4); the minimum is -8 when (x,y) = (-4, -4). If you
plot the hyperbola, you\'ll find the the line x + y = -8 is tangent to the graph at the vertex in the
third quadrant..
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find the minimum value of x+y, subject to the constraints xy=16, x0.pdf
1. find the minimum value of x+y, subject to the constraints xy=16, x>0, y>0
Solution
Let f(x,y) = x + y and g(x, y) = xy. ?f = <1, 1> and ?g = . The system of equations
to solve is ?f = ??g, g = 16. The first two equations give x = y = 1/?. Plugging this into the
constraint gives 1/?² = 16 ==> ? = ±1/4 ==> x = y = ±4. There are two optimal values, one max
and one min. The max is 8 when (x,y) = (4, 4); the minimum is -8 when (x,y) = (-4, -4). If you
plot the hyperbola, you'll find the the line x + y = -8 is tangent to the graph at the vertex in the
third quadrant.
