find the stationary point of the function
z=4x^2+10xy+4y^2-x^2y^2
and also determine their nature
plz show me step i don't know it
You need both partials to be zero. That is,
#1: 3y^3 + 2xy = 0
#2: 3(3x+1)y^2 + x^2 = 0
From #1, we get x = -3y^2/2
Plug that into #2 and you get y^2 = 4/15
So, z has stationary points at (-2/5, ±2/√15)
To find the stationary points of the given function z=4x^2+10xy+4y^2-x^2y^2, we need to solve the partial derivative equations with respect to x and y and set them equal to zero.
Step 1: Find the partial derivative with respect to x (denoted as ∂z/∂x):
∂z/∂x = 8x + 10y - 2xy^2
Step 2: Set the above derivative equation equal to zero and solve for x:
8x + 10y - 2xy^2 = 0
Step 3: Find the partial derivative with respect to y (denoted as ∂z/∂y):
∂z/∂y = 10x + 8y - 2x^2y
Step 4: Set the above derivative equation equal to zero and solve for y:
10x + 8y - 2x^2y = 0
Step 5: Now we have a system of two equations with two variables:
8x + 10y - 2xy^2 = 0 (Equation 1)
10x + 8y - 2x^2y = 0 (Equation 2)
Step 6: Solve the system of equations to find the values of x and y that satisfy both equations simultaneously.
There are multiple methods to solve this system, such as substitution or elimination. For simplicity, we'll use the substitution method.
Solve Equation 1 for x:
8x + 10y - 2xy^2 = 0
x = (2xy^2 - 10y)/8
Substitute this into Equation 2:
10(2xy^2 - 10y)/8 + 8y - 2(2xy^2 - 10y)^2y = 0
Simplify and solve for y.
To find the stationary points of the given function z = 4x^2 + 10xy + 4y^2 - x^2y^2, we need to take partial derivatives with respect to x and y and set them equal to zero.
Step 1: Find the partial derivative with respect to x.
To find the partial derivative of the function with respect to x (denoted as dz/dx), you differentiate the function with respect to x while treating y as a constant.
Taking the derivative of each term separately:
∂z/∂x = ∂(4x^2)/∂x + ∂(10xy)/∂x + ∂(4y^2)/∂x - ∂(x^2y^2)/∂x
Simplifying:
∂z/∂x = 8x + 10y - 2xy^2
Step 2: Find the partial derivative with respect to y.
Similarly, to find the partial derivative of the function with respect to y (denoted as dz/dy), you differentiate the function with respect to y while treating x as a constant.
Taking the derivative of each term separately:
∂z/∂y = ∂(4x^2)/∂y + ∂(10xy)/∂y + ∂(4y^2)/∂y - ∂(x^2y^2)/∂y
Simplifying:
∂z/∂y = 0 + 10x + 8y - 2x^2y
Step 3: Set the partial derivatives equal to zero.
Setting both ∂z/∂x and ∂z/∂y equal to zero will give us the x and y values of the stationary points.
∂z/∂x = 8x + 10y - 2xy^2 = 0
∂z/∂y = 10x + 8y - 2x^2y = 0
Step 4: Solve the system of equations.
You can solve these two equations simultaneously to find the values of x and y.
From ∂z/∂x = 8x + 10y - 2xy^2 = 0, you can isolate x:
8x = 2xy^2 - 10y
4x = xy^2 - 5y
And from ∂z/∂y = 10x + 8y - 2x^2y = 0, you can isolate x:
10x = 2x^2y - 8y
5x = x^2y - 4y
Substituting the expression for x from the first equation into the second equation:
5(xy^2 - 5y) = (xy^2 - 5y)^2 - 4y
5xy^2 - 25y = (xy^2 - 10y + 25y^2) - 4y
5xy^2 - 25y = xy^2 - 10y + 25y^2 - 4y
5xy^2 - xy^2 - 25y + 10y - 25y^2 + 4y = 0
4xy^2 - xy^2 - 11y + 4y - 25y^2 = 0
3xy^2 - 7y - 25y^2 = 0
At this point, the equations become quite complex, and it may not be possible to find exact solutions analytically. In such cases, numerical methods or graphical approach can be utilized.