Maxima and Minima for a two variable function

In summary, the function f(x,y) = (sin x)(cos y) has infinitely many critical points located at x = pi/2, -pi/2, 3pi/2, -3pi/2, ..., and y = kpi, where k is any integer.
  • #1
Odyssey
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Greetings, can you guys please help me with my assignment??

I am supposed to find and classify all critical points of the function f(x,y) = (sin x)(cos y)

Now I took the first partials with repsect to x and y and they are (cos x)(cos y) = 0 and (-sin x)(sin y) = 0, respectively.

Now I know fx = 0 when either cos x = 0 or cos y = 0 and cos x = 0 when x is pi/2, -pi/2, 3pi/2, -3pi/2, ...

and fy = 0 when either sin x = 0 or sin y = 0 and sin x = 0 when x = kpi, k being integers...

But I don't know if there are infinitely many critical points or no critical points?? If there are infinitely many, where are they at??

Thanks for the help! :tongue2:
 
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  • #2
There are critical points as you showed, and there are infinitely many of them as you know from a sin(x) or cos(x) graph.
 

1. What is a maximum/minimum value for a two variable function?

A maximum value for a two variable function is the highest point on the surface of the function, while a minimum value is the lowest point. These points represent the highest and lowest possible values for the function.

2. How do you find the maximum/minimum value for a two variable function?

To find the maximum or minimum value for a two variable function, you must first take the partial derivatives with respect to each variable and set them equal to zero. Then, solve the resulting system of equations to find the critical points. The highest/lowest point among the critical points is the maximum/minimum value.

3. What is a saddle point in a two variable function?

A saddle point in a two variable function is a point where the surface of the function curves downward in one direction and upward in the other. It is neither a maximum nor a minimum point, but rather a point of inflection.

4. How do you determine if a critical point is a maximum, minimum, or saddle point?

To determine the type of critical point, you can use the second derivative test. If the second derivative is positive at the critical point, it is a minimum. If the second derivative is negative, it is a maximum. If the second derivative is zero, you must use further analysis to determine if it is a saddle point.

5. Can a two variable function have multiple maximum/minimum values?

Yes, a two variable function can have multiple maximum/minimum values. This occurs when there are multiple critical points that have the same function value. In this case, all of the maximum/minimum values must be considered when analyzing the function.

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