let f(x,y)=Y^2+xy-x^2-5y+2. find the critical point
To find the critical points of the function f(x, y) = y^2 + xy - x^2 - 5y + 2, we need to find the values of x and y where the partial derivatives are equal to zero.
Step 1: Calculate the partial derivative with respect to x.
To find ∂f/∂x, we differentiate f(x, y) with respect to x, treating y as a constant:
∂f/∂x = (∂/∂x) (y^2 + xy - x^2 - 5y + 2)
Differentiating each term gives us:
∂f/∂x = 0 + y - 2x + 0 - 0 = y - 2x
Step 2: Calculate the partial derivative with respect to y.
To find ∂f/∂y, we differentiate f(x, y) with respect to y, treating x as a constant:
∂f/∂y = (∂/∂y) (y^2 + xy - x^2 - 5y + 2)
Differentiating each term gives us:
∂f/∂y = 2y + x - 5 -0 + 0 = 2y + x - 5
Step 3: Set the partial derivatives equal to zero and solve the system of equations.
Setting ∂f/∂x = 0 and ∂f/∂y = 0, we have the following equations:
y - 2x = 0 ---(1)
2y + x - 5 = 0 ---(2)
From equation (1), we can isolate y:
y = 2x
Substituting this into equation (2), we get:
2(2x) + x - 5 = 0
4x + x - 5 = 0
5x = 5
x = 1
Now, substitute the value of x into equation (1) to find y:
y - 2(1) = 0
y - 2 = 0
y = 2
Therefore, the critical point of the function f(x, y) is (1, 2).