A company manufactures two types of toasters. The total revenue from x units of type 1 and y...

Question:

A company manufactures two types of toasters. The total revenue from x units of type 1 and y units of type 2 is {eq}R = 25x +25y-3xy-2x^2 - y^2 {/eq}. Find x and y so as to maximize the revenue.

Sufficient conditions for a function to have a minimum/maximum:

Let a function {eq}f(x,y) {/eq} be continuous and possess first and second order partial derivatives at a point {eq}P(a,b). {/eq} If {eq}P(a,b) {/eq} is a critical point, then the point {eq}P {/eq} is a point of

• relative minimum if {eq}rt - s^2 > 0 \ \ \text{and} \ \ r > 0. {/eq}
• relative maximum if {eq}rt - s^2 > 0 \ \ \text{and} \ \ r < 0. {/eq}

where {eq}r = f_{xx}(a,b) , \ \ s = f_{xy}(a,b) \ \ \text{and} \ \ t = f_{yy}(a,b). {/eq}

The given revenue function is:

{eq}\hspace{30mm} \displaystyle{ R(x,y) = 25x +25y-3xy-2x^2 - y^2 \\ R_{x} = 25 - 3y - 4x \\ R_{y} = 25 - 3x - 2y } {/eq}

To find critical points solve the system of equation:

{eq}\hspace{30mm} \displaystyle{ 25 - 3y - 4x = 0 \\ 25 - 3x - 2y = 0 } {/eq}

The solution of this system is {eq}x = 25 \ \ \text{and} \ \ y = -25. {/eq} Therefore the critical point is {eq}(25, -25). {/eq}

Now

{eq}\hspace{30mm} \displaystyle{ R_{xx} = - 4 \\ R_{yy} = - 2 \\ R_{xy} = -3 } {/eq}

Therefore the discriminant is given by:

{eq}\hspace{30mm} \displaystyle{ D = R_{xx} R_{yy} - R_{xy}^2 \\ D = - 4 \cdot -2 - (-3)^2 \\ D = 8 - 9 \\ D = -1 } {/eq}

Hence the given revenue function does not have any maximum or minimum at the obtained critical point.