What is the Mistake in Finding the Critical Point for f(x,y)=xy+(144/x)+(12/y)?

In summary, using partial derivatives and setting them equal to zero, we can find the critical points of a function. In this conversation, the function f(x,y) has a relative minimum at (0,0). However, we must also consider the second order partial derivatives to determine if this is a true extremum.
  • #1
MathNoob123
15
0

Homework Statement


f(x,y)=xy+(144/x)+(12/y)
f has a relative minimum at ( , , )

Homework Equations



Partial derivatives

The Attempt at a Solution


fx=y-(144/x^2)
fy=x-(12/y^2)

I setted these to zero. Multiplies both sides so there is no fraction.

Came out to be something like~~~ x2y-144=0
xy2-12=0

then did the factoring stuff and got

x2-12xy2=0

This tells me that the critical point is (0,0), but this is the wrong answer.

PLEASE TELL ME WHAT I AM DOING WRONG.

THANK YOU SO MUCH TO THOSE WHO HELP. THIS FORUM IS HELPING ME BECOME A BETTER MATHMATICIAN AND THEREFORE, MY TEST GRADES ARE GOING UP. THANK YOU.
 
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  • #2
Your partial derivatives and general procedure are correct:smile:; but your attempt to solve for x and y is very confusing...try solving the f_y=0 equation for x, and substitute that result into your f_x=0 equation and solve for y...
 
  • #3
just to add - there can be (and is in this case) more than one stationary point

as you mentioned (0,0) is a stationary point, but that doesn't automatically make it an extremum - need to look at the 2nd order partial dervatievs to determine that
Edit - (0,0) is not a critical point
 
Last edited:
  • #4
(0, 0) most certainly is NOT a stationary point- it's not even a point on the graph. f is not defined at (0,0).
And I get only one critical point.
 
  • #5
yeah - looked at that a little too quickly, sorry for the mis-steer
 

FAQ: What is the Mistake in Finding the Critical Point for f(x,y)=xy+(144/x)+(12/y)?

What is the purpose of classifying critical points?

The purpose of classifying critical points is to determine the behavior of a function at that point. It helps to identify whether the point is a maximum, minimum, or saddle point.

How do you determine the type of critical point?

The type of critical point can be determined by analyzing the second derivative of the function at that point. If the second derivative is positive, the point is a minimum. If the second derivative is negative, the point is a maximum. If the second derivative is zero, further analysis is needed to determine if the point is a minimum, maximum, or saddle point.

Can a function have more than one critical point?

Yes, a function can have multiple critical points. These points can be classified as maximum, minimum, or saddle points individually.

Is it necessary to classify critical points?

Classifying critical points is not always necessary, but it can provide valuable information about the behavior of a function. It can also help solve optimization problems by identifying the maximum or minimum value of a function.

How are critical points related to the graph of a function?

Critical points are points on the graph of a function where the slope is zero or undefined. They are important because they can indicate the location of maximum, minimum, or saddle points on the graph.

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