The absolute maximum of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 x$ on or inside the rectangle with vertices (0, -3), (5, -3), (5, 3), and (0, 3) is

A)19
B)16
C)14
D)12
E)8

Question

The absolute maximum of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 x$ on or inside the rectangle with vertices (0, -3), (5, -3), (5, 3), and (0, 3) is

A)19
B)16
C)14
D)12
E)8

Quizplus · Accepted Answer

The function is quadratic and opens upwards, so the maximum occurs on the boundary of the rectangle. Evaluating $f(x, y)$ at the vertices, the maximum value is 19 at (5, 3).

The Absolute Maximum Of f(x,y)=x2+y2+xy−6xf ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 xf(x,y)=x2+y2+xy−6x

The Absolute Maximum Of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 x$