Stationary Points: Finding Stationary Points of f(x,y)

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In summary, the stationary points of the function f(x,y) = (x^2 + 1/2(xy) + y^2)e^(x+y) can be found by solving the equations df/dx = 0 and df/dy = 0 simultaneously. After substituting y = x, the resulting equation can be solved using the quadratic formula to find the values of x that make the first equation equal to 0. These values can then be substituted back into either equation to find the corresponding y-values.
  • #1
gtfitzpatrick
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Homework Statement


determine the stationary points of the function f(x,y) = (x^2 + 1/2(xy) + y^2)e^(x+y)


Homework Equations





The Attempt at a Solution


first i got

df/dx= (x^2+(1/2)xy+y^2)e^(x+y)+(2x+(1/2)y)e^(x+y)

df/dy= (x^2+(1/2)xy+y^2)e^(x+y)+((1/2)x+2y)e^(x+y)

i then let them both = 0

and i get

(x^2+2x+(1/2)xy+(1/2)y+y^2)=0

and

(x^2+(1/2)x+(1/2)xy+2y+y^2)=0

i've tried different to solve them simultaniously and then sub back in. I've tried to factorise them which i think i should do but can't seem to get it could anyone give me some pointers please?
 
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  • #2
Using the quadratic formula you can solve df/dx = 0 as y [itex]\in[/itex] {1/4 (-1 - Sqrt[1 - 32 x - 16 x^2 - 8 xy]), 1/4 (-1 + Sqrt[1 - 32 x - 16 x^2 - 8 xy])}, then sub into df/dy = 0.
 
Last edited:
  • #3
gtfitzpatrick said:

Homework Statement


determine the stationary points of the function f(x,y) = (x^2 + 1/2(xy) + y^2)e^(x+y)


Homework Equations





The Attempt at a Solution


first i got

df/dx= (x^2+(1/2)xy+y^2)e^(x+y)+(2x+(1/2)y)e^(x+y)

df/dy= (x^2+(1/2)xy+y^2)e^(x+y)+((1/2)x+2y)e^(x+y)

i then let them both = 0

and i get

(x^2+2x+(1/2)xy+(1/2)y+y^2)=0

and

(x^2+(1/2)x+(1/2)xy+2y+y^2)=0
Subtracting the second from the first gives you (3/2)x- (3/2)y= 0 or y= x. Put that back into either equation (because of the symmetry) gives you [itex]x^2+ 2x+ (1/2)x^2+ (1/2)x+ x^2= (5/2)x^2+ (5/2)x= 0[/itex] or x(x+ 1)= 0.

i've tried different to solve them simultaniously and then sub back in. I've tried to factorise them which i think i should do but can't seem to get it could anyone give me some pointers please?
 
  • #4
gtfitzpatrick said:

Homework Statement


determine the stationary points of the function f(x,y) = (x^2 + 1/2(xy) + y^2)e^(x+y)


Homework Equations





The Attempt at a Solution


first i got

df/dx= (x^2+(1/2)xy+y^2)e^(x+y)+(2x+(1/2)y)e^(x+y)

df/dy= (x^2+(1/2)xy+y^2)e^(x+y)+((1/2)x+2y)e^(x+y)

i then let them both = 0

and i get

(x^2+2x+(1/2)xy+(1/2)y+y^2)=0

and

(x^2+(1/2)x+(1/2)xy+2y+y^2)=0
Subtracting the second from the first gives you (3/2)x- (3/2)y= 0 or y= x. Put that back into either equation (because of the symmetry) gives you [itex]x^2+ 2x+ (1/2)x^2+ (1/2)x+ x^2= (5/2)x^2+ (5/2)x= 0 or x(x+ 1)= 0[/itex].

i've tried different to solve them simultaniously and then sub back in. I've tried to factorise them which i think i should do but can't seem to get it could anyone give me some pointers please?
 

FAQ: Stationary Points: Finding Stationary Points of f(x,y)

1. What is a stationary point?

A stationary point is a point on a graph where the gradient is equal to zero, meaning that the slope of the curve is neither increasing nor decreasing at that point.

2. Why is it important to find stationary points?

Stationary points are important because they can help us identify the maximum and minimum values of a function. This is useful in optimization problems where we want to find the best possible solution.

3. How do you find stationary points?

To find stationary points, we need to take the partial derivatives of the function with respect to each variable and set them equal to zero. Then, we can solve the resulting system of equations to find the coordinates of the stationary points.

4. What do the different types of stationary points represent?

There are three types of stationary points: maximum, minimum, and saddle points. A maximum point is where the function has the highest value, a minimum point is where the function has the lowest value, and a saddle point is a point where the function changes from increasing to decreasing or vice versa.

5. Can a function have more than one stationary point?

Yes, a function can have multiple stationary points. In fact, most functions have more than one stationary point. It is important to check all potential stationary points to determine which one is the maximum, minimum, or saddle point.

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