Maxima and minima of functions of two variables

In summary, to locate all relative maxima, relative minima, and saddle points of the function f(x,y)=ysinx, we first find the critical points, which are at (0,0), (∏,0), (2∏,0), and so on. Then, we find the second order partial derivatives fxx(x,y)=-ysinx and fyy(x,y)=cosx. Finally, we use the formula fxx*fyy-(fxy)2 to determine whether there are any relative extrema. In this case, since fxx*fyy-(fxy)2=-1, there is a saddle point at the critical points.
  • #1
Mdhiggenz
327
1

Homework Statement



Locate all relative maxima, relative minima ,and saddle points if any.

f(x,y)=ysinx

fx(x,y)=ycosx

fy(x,y)=sinx

ycosx=0 sinx=0
y=0 x=0,∏,2∏... up until infinity
Critical points at (0,0),(∏,0),(2∏,0)...

fxx(x,y)=-ysinx

fyy(x,y)= cosx

fxx*fyy-f(x,y)2→0-0=0 ∴ no relative extrema,

however the book says that fxx*fyy-f(x,y)2=-1 which means saddle point.

I don't understand if y is = to zero.. the function ysinx will always be zero..

Homework Equations





The Attempt at a Solution

 
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  • #2
Fyy=0
 
  • #3
Sorry that was a typo, but still doesn't help regardless.
 
  • #4
Mdhiggenz said:

Homework Statement



Locate all relative maxima, relative minima ,and saddle points if any.

f(x,y)=ysinx

fx(x,y)=ycosx

fy(x,y)=sinx

ycosx=0 sinx=0
y=0 x=0,∏,2∏... up until infinity
Critical points at (0,0),(∏,0),(2∏,0)...

fxx(x,y)=-ysinx

fyy(x,y)= cosx

fxx*fyy-f(x,y)2→0-0=0 ∴ no relative extrema,

however the book says that fxx*fyy-f(x,y)2=-1 which means saddle point.

I don't understand if y is = to zero.. the function ysinx will always be zero..

Homework Equations



The Attempt at a Solution

Isn't fy(x,y) = sin(x) ?

Then what is fyy(x,y) ? Isn't it zero?

Also, What is fxy(x,y) ?

You need fxx*fyy-(fxy)2 , not fxx*fyy-f(x,y)2
 
Last edited:

Related to Maxima and minima of functions of two variables

1. What is a maxima and minima of a function of two variables?

A maxima and minima of a function of two variables are the highest and lowest values that the function can take within a given range of two independent variables. They represent the peak and valley points of the function's graph.

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

To find the maxima and minima of a function of two variables, you can use the method of partial derivatives. This involves finding the partial derivatives of the function with respect to each variable, setting them equal to 0, and solving for the variables. The resulting values will give you the coordinates of the maxima and minima points.

3. Can a function of two variables have more than one maxima or minima?

Yes, a function of two variables can have multiple maxima and minima. This can happen when the function has multiple peaks or valleys within the given range of variables. These points are known as local maxima and minima, while the highest or lowest point in the entire range is known as the global maxima or minima.

4. How do you determine if a maxima or minima is a maximum or minimum point?

To determine whether a maxima or minima point is a maximum or minimum, you can use the second derivative test. This involves taking the second partial derivatives of the function and evaluating them at the coordinates of the maxima or minima point. If the value is positive, it is a minimum point, and if it is negative, it is a maximum point.

5. How are maxima and minima of a function of two variables used in real-world applications?

Maxima and minima of a function of two variables are commonly used in optimization problems. For example, in economics, these points can represent the maximum profit or minimum cost for a certain production level. In engineering, they can represent the maximum strength or minimum weight for a given design. They are also used in data analysis to identify the peak and valley points in a dataset.

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