How do I solve systems of equations to find local max, min, and saddle points?

In summary, a maxima, minima, saddle point are the three types of critical points on a graph of a function. To find them, you can take the derivative of the function and set it equal to 0. The first derivative test tells us whether a critical point is a maxima, minima, or saddle point. A function can have multiple maxima and minima, which can be either local or global. In optimization problems, these critical points are crucial as they represent the highest and lowest values of a function and help determine the optimal solution.
  • #1
says
594
12

Homework Statement


Find the local max, min, and saddle point for the function:
f(x,y) = 2x^2+3xy+4y^2-5x+2y

Homework Equations

The Attempt at a Solution


I've taken the two partial derivatives

Fx = 4x + 3y - 5
Fy = 3x + 8y + 2

I know that the critical points will sit where both of theses partial derivatives = 0
i.e.

Fx = 4x + 3y - 5 = 0
Fy = 3x + 8y + 2 = 0

The problem I have here though is that I don't know how to solve the system of equations.

I know once I've solved the system of equations I can use the determinant of the jacobian matrix to see whether they are local max, min, or saddle points...

Any help with solving the system of equations would be much appreciated. I've had a bit of trouble solving systems of equations in the past.
 
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  • #2
says said:

Homework Statement


Find the local max, min, and saddle point for the function:
f(x,y) = 2x^2+3xy+4y^2-5x+2y

Homework Equations

The Attempt at a Solution


I've taken the two partial derivatives

Fx = 4x + 3y - 5
Fy = 3x + 8y + 2

I know that the critical points will sit where both of theses partial derivatives = 0
i.e.

Fx = 4x + 3y - 5 = 0
Fy = 3x + 8y + 2 = 0

The problem I have here though is that I don't know how to solve the system of equations.

Really? You never solved a set of simultaneous linear equations in your algebra courses?

You can use Cramer's Rule or elimination to solve the system above.

http://www.coolmath.com/algebra/14-determinants-cramers-rule/01-determinants-cramers-rule-2x2-01

http://www.purplemath.com/modules/systlin6.htm

I know once I've solved the system of equations I can use the determinant of the jacobian matrix to see whether they are local max, min, or saddle points...

Any help with solving the system of equations would be much appreciated. I've had a bit of trouble solving systems of equations in the past.
 
  • #3
Generally speaking, it is not a good idea to try to learn Calculus until after you have a firm grasp of algebra. You have the equations
Fx = 4x + 3y - 5 = 0 and Fy = 3x + 8y + 2 = 0. If you multiply the first equation by 3 you have 12x+ 9y- 15= 0. If you multiply the second equation by 4 you have 12x+ 32y+ 8= 0. Now the x term in each equation has the same coefficient so subtracting one equation from the other eliminates x and you have a single equation to solve for y.
 

1. What is a maxima, minima, saddle point?

A maxima, minima, saddle point refers to the three types of critical points on a graph of a function. A maxima is the highest point on the graph, a minima is the lowest point on the graph, and a saddle point is a point that is neither a maxima nor a minima, but is a point of inflection.

2. How do you find the maxima and minima of a function?

To find the maxima and minima of a function, you can take the derivative of the function and set it equal to 0. Then, solve for the x-values where the derivative is equal to 0. These x-values will correspond to the critical points, and by plugging them back into the original function, you can find the corresponding y-values for the maxima and minima.

3. What does the first derivative test tell us about critical points?

The first derivative test tells us whether a critical point is a maxima, minima, or saddle point. If the derivative is positive at a critical point, then the point is a local minima. If the derivative is negative at a critical point, then the point is a local maxima. If the derivative is 0 at a critical point, then further analysis is needed to determine the type of point.

4. Can a function have more than one maxima or minima?

Yes, a function can have multiple maxima and minima. These points can be either local or global, where local maxima and minima are the highest or lowest points in a specific interval, and global maxima and minima are the highest or lowest points in the entire domain of the function.

5. How do maxima and minima relate to optimization problems?

Maxima and minima are essential in optimization problems as they represent the highest and lowest values of a function. In optimization problems, we are trying to find the maximum or minimum value of a function, which can be found at the maxima or minima of the function. By finding these critical points, we can determine the optimal solution to an optimization problem.

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