A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from units of running shoes and units of basketball shoes is: $R = - 6 x _ { 1 } ^ { 2 } - 9 x _ { 2 } ^ { 2 } - 2 x _ { 1 } x _ { 2 } + 37 x _ { 1 } + 95 x _ { 2 }$
where $x _ { 1 }$ and $x _ { 2 }$ are in thousands of units. Find $x _ { 1 }$ and $x _ { 2 }$ so as to maximize the revenue.
A) $x _ { 1 } = \frac { 119 } { 53 } , x _ { 2 } = \frac { 533 } { 106 }$
B) $x _ { 1 } = \frac { 533 } { 106 } , x _ { 2 } = \frac { 119 } { 53 }$
C) $x _ { 1 } = \frac { 533 } { 106 } , x _ { 2 } = \frac { 119 } { 106 }$
D) $x _ { 1 } = \frac { 238 } { 107 } , x _ { 2 } = \frac { 533 } { 107 }$
E) $x _ { 1 } = \frac { 533 } { 107 } , x _ { 2 } = \frac { 238 } { 107 }$

Question

Quizplus · Accepted Answer

To maximize the revenue, we need to find the critical points of the revenue function $R = - 6 x _ { 1 } ^ { 2 } - 9 x _ { 2 } ^ { 2 } - 2 x _ { 1 } x _ { 2 } + 37 x _ { 1 } + 95 x _ { 2 }$ by taking the partial derivatives with respect to $x_1$ and $x_2$, setting them equal to zero, and solving the resulting system of equations. This process will yield the values of $x_1$ and $x_2$ that maximize revenue, which corresponds to choice A.

A Company Manufactures Two Types of Sneakers: Running Shoes and Basketball