# Find all relative maxima/minima and saddle points

In summary, the problem involves finding critical points of the function f(x,y)=2x^3+xy^2+5x^2+y^2. By taking the partial derivatives, we get fx(x,y)=6x^2+y^2+25x and fy(x,y)=2xy+2y. Setting both of these equal to 0, we can find the critical points at y=0 and x=-1. However, it is important to double check the arithmetic for fx before proceeding with the solution.

## Homework Statement

f(x,y)=2x^3+xy^2+5x^2+y^2

## The Attempt at a Solution

fx(x,y)=6x^2+y^2+25x
fy(x,y)=2xy+2y
fxx(x,y)=12x+25
fyy(x,y)=2y+2
fxy(x,y)=2y
I found the partial derivatives from the equation. I am stuck at finding the critical points from the first two fx(x,y) and fy(x,y). I equaled 0=6x^2+y^2+25x and solved for y^2. I then got y^2=-6x^2-25x and tried substituting it in, fy(x,y)= 2x(6-x^2-25x)^2+2(-6x^2-25x)^2. which did not work out well as I was able to do in similar problems.

## Homework Statement

f(x,y)=2x^3+xy^2+5x^2+y^2

## The Attempt at a Solution

fx(x,y)=6x^2+y^2+25x
fy(x,y)=2xy+2y
fxx(x,y)=12x+25
fyy(x,y)=2y+2
fxy(x,y)=2y
I found the partial derivatives from the equation. I am stuck at finding the critical points from the first two fx(x,y) and fy(x,y). I equaled 0=6x^2+y^2+25x and solved for y^2. I then got y^2=-6x^2-25x and tried substituting it in, fy(x,y)= 2x(6-x^2-25x)^2+2(-6x^2-25x)^2. which did not work out well as I was able to do in similar problems.
You have ##f_y = 2xy+2y##. Set that equal to ##0##, and factor it getting ##2y(x+1)=0##. That tells you there are two possibilities, either ##y=0## or ##x=-1##. Go from there.

LCKurtz said:
You have ##f_y = 2xy+2y##. Set that equal to ##0##, and factor it getting ##2y(x+1)=0##. That tells you there are two possibilities, either ##y=0## or ##x=-1##. Go from there.

that was a mistype on the fx either than that thank you.

## 1. What is a relative maximum/minimum?

A relative maximum/minimum is a point on a graph where the function reaches its highest/lowest value in a specific interval, but not necessarily the highest/lowest value in the entire domain.

## 2. How do you find relative maxima/minima?

To find relative maxima/minima, you can use the first derivative test or the second derivative test. The first derivative test involves finding the critical points of the function and determining if they are points of minima or maxima. The second derivative test involves finding the points where the second derivative of the function is equal to zero, and then using the concavity of the function to determine if they are points of minima or maxima.

## 3. What is a saddle point?

A saddle point is a point on a graph where the function has a critical point, but it is neither a relative maximum nor a relative minimum. Instead, it is a point of inflection where the function changes from concave up to concave down or vice versa.

## 4. How do you find saddle points?

To find saddle points, you can use the second derivative test. If the second derivative of the function is equal to zero at a certain point, and the concavity changes at that point, then it is a saddle point.

## 5. Why is finding relative maxima/minima and saddle points important?

Finding relative maxima/minima and saddle points is important because it helps us understand the behavior of a function and identify important points on a graph. These points can also be used to optimize a function and solve real-world problems in various fields such as economics, physics, and engineering.

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