Question

find the derivatives $r_{xx}$, $r_{yy}$, and $r_{xy}$. $r_{xx}=-8$, $r_{yy}=-18$, $r_{xy}=-8$ selling $square$ spas and $square$ solar heaters gives the maximum revenue of $$$$square$. (simplify your answers.)

Explanation:

Step1: Find \(R_{xx}\)

Differentiate \(R(x,y) = 12 + 208x+308y - 4x^{2}-9y^{2}-8xy\) with respect to \(x\) twice.
First - derivative: \(R_{x}=\frac{\partial R}{\partial x}=208 - 8x-8y\).
Second - derivative: \(R_{xx}=\frac{\partial^{2}R}{\partial x^{2}}=-8\).

Step2: Find \(R_{yy}\)

Differentiate \(R(x,y)\) with respect to \(y\) twice.
First - derivative: \(R_{y}=\frac{\partial R}{\partial y}=308 - 18y-8x\).
Second - derivative: \(R_{yy}=\frac{\partial^{2}R}{\partial y^{2}}=-18\).

Step3: Find \(R_{xy}\)

Differentiate \(R_{x}\) with respect to \(y\).
Since \(R_{x}=208 - 8x - 8y\), then \(R_{xy}=\frac{\partial R_{x}}{\partial y}=-8\).

Step4: Use the second - derivative test for maxima

The second - derivative test for a function \(z = R(x,y)\) at a critical point \((a,b)\) uses the discriminant \(D=R_{xx}(a,b)R_{yy}(a,b)-[R_{xy}(a,b)]^{2}\).
First, find the critical points by setting \(R_{x}=0\) and \(R_{y}=0\).
\(R_{x}=208 - 8x-8y = 0\) implies \(x + y=26\) (divide by 8).
\(R_{y}=308 - 18y-8x = 0\). Substitute \(x = 26 - y\) into \(R_{y}\):

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If \(y = 10\), then \(x=26 - 10=16\).
Now, \(D=(-8)\times(-18)-(-8)^{2}=144 - 64 = 80>0\) and \(R_{xx}=-8<0\). So, the function \(R(x,y)\) has a maximum at \((x = 16,y = 10)\).

Step5: Calculate the maximum revenue

Substitute \(x = 16\) and \(y = 10\) into \(R(x,y)\):

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Answer:

\(R_{xx}=-8\), \(R_{yy}=-18\), \(R_{xy}=-8\), and the maximum revenue from selling spas and solar - heaters is \(\$3216\).