Find critical points for
f(x,y)= x^3+y^3-15x^2-6xy+18y^2+39x+138y
I know you first take the partial derivative of x and y, but I am not sure where to go from there.
critical points are where they are both zero. So, you need
x^3+y^3-15x^2-6xy+18y^2+39x+138y
3x^2-30x-6y+39 = 0
3y^2-6x+36y+138 = 0
http://www.wolframalpha.com/input/?i=solve+3x%5E2-30x-6y%2B39+%3D+0,+3y%5E2-6x%2B36y%2B138+%3D+0
http://www.wolframalpha.com/input/?i=minima+x%5E3%2By%5E3-15x%5E2-6xy%2B18y%5E2%2B39x%2B138y
To find the critical points of a function, you need to set the partial derivatives equal to zero and solve the resulting system of equations. Let's start by finding the partial derivatives of the function f(x, y) with respect to x and y:
∂f/∂x = 3x^2 - 30x - 6y + 39
∂f/∂y = 3y^2 - 6x + 36y + 138
Now, we can set these partial derivatives equal to zero and solve the system:
3x^2 - 30x - 6y + 39 = 0 ----------- (Equation 1)
3y^2 - 6x + 36y + 138 = 0 ----------- (Equation 2)
To solve this system of equations, we can use various methods such as substitution or elimination. However, in this case, let's solve Equation 2 for x and substitute it into Equation 1:
From Equation 2:
6x = 3y^2 + 36y + 138
x = (3y^2 + 36y + 138)/6
x = (y^2 + 12y + 46)/2
Substituting this value of x into Equation 1:
3(y^2 + 12y + 46)^2/4 - 30(y^2 + 12y + 46)/2 - 6y + 39 = 0
Multiply through by 4 to eliminate the fractions:
3(y^2 + 12y + 46)^2 - 60(y^2 + 12y + 46) - 12y + 156 = 0
Expanding and simplifying:
3y^4 + 72y^3 + 678y^2 + 2436y + 2076 - 60y^2 - 720y - 2760 - 12y + 156 = 0
Combining like terms:
3y^4 + 72y^3 + 618y^2 - 996y + 372 = 0
Now, to find the critical points, we need to solve this equation. However, the solution involves finding the roots of a quartic polynomial, which is a complex process. Typically, numerical methods or software are used to find the approximate solutions.
In this case, if you have access to numerical software or graphing calculators, you can input the equation into those tools to find the approximate critical points.
Alternatively, you can use computer programming languages such as Python to solve for the roots numerically. There are libraries such as NumPy or SymPy that can assist in finding the solutions.
For example, using Python with NumPy:
import numpy as np
coefs = [3, 72, 618, -996, 372]
roots = np.roots(coefs)
print(roots)
Running this code would give you the approximate values of the roots, which correspond to the critical points of the function.