Question

Use calculus to find the first order conditions. Use solver then to solve the first order conditions. Show your work for the first order conditions and include the relevant portion of excel. f(x,y)=-22x^2+22xy-11y^2+110x-40y-23

Answer 1

The values of x and y are 70/22 and 30/22 respectively by using the first-order condition of differential calculus.

What is the first-order condition in differential calculus?

A first-order differential equation is represented by the equation $\mathbf{ \dfrac{dy}{dx} =f (x,y) }$with 2 variables x & y, including its function f(x,y) specified on a xy-plane.

Given that:

$\mathbf{f(x,y) =-22x^2+22xy-11y^2+110x-40y-23}$

Let us first differentiate the above equation with respect to x, we have:

$\mathbf{\dfrac{\partial f(x,y) }{\partial x} = -44x +22y -0+110-0-0=0}$

$\mathbf{\implies -44x +22y+110=0}$        (multiply by -1)

44x - 22y = 110    ------ (equation 1)

Now, differentiating with respect to y, we have:

$\mathbf{\dfrac{\partial f(x,y) }{\partial y} =0 +22x-22y +0-40-0=0}$

$\mathbf{\implies 22x-22y -40=0}$

22x - 22y = 40      ----- (equation 2)

Now, we have a system of equations:

44x - 22y = 110

-                      ---- ( subtracting equation 2 from 1; elimination method)

22x - 22y = 40  

22x    + 0  = 70    

x = 70/22

Replacing the value of x into equation (1), we have:

44x - 22y = 110

44(70/22) - 22y = 110

140 - 22y = 110

140 - 110 = 22y

30 = 22y

y = 30/22

Learn more about the first-order conditions in differential calculus here;

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