Find the maximum value of the function $f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160$
A)90
B)160
C)50
D)15
E)80
F)135
G)16
H)5

Question

Find the maximum value of the function $f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160$ 
A)90
B)160
C)50
D)15
E)80
F)135
G)16
H)5

Quizplus · Accepted Answer

The function is a quadratic with negative coefficients for the squared terms, so it has a maximum value. The maximum value occurs at the vertex of the parabola, which is given by the formula $ ( - b / 2a , - c / 2b ) $. In this case, $ a = -1 $, $ b = 30 $, and $ c = -160 $, so the vertex is $ ( -30 / 2(-1) , -(-160) / 2(30) ) = (15, 8) $. The maximum value of the function is $ f(15, 8) = 10(15) + 30(8) - (15)^2 - (8)^2 - 160 = 90 $.

Find the Maximum Value of the Function f(x,y)=10x+30y−x2−y2−160f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160f(x,y)=10x+30y−x2−y2−160

Find the Maximum Value of the Function $f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160$