Critical Points & Extrema of Multivariable Function

In summary, the function f(x,y)=x^(2/3)+y^(2/3) has a critical point at (0,0) where f_x and f_y are both equal to 0. However, the Second Partials Test fails at this point, indicating that it is not a relative extremum. This is because the determinant d is equal to 0, meaning that the Hessian matrix is not invertible. The textbook states that there is an absolute minimum at this point, which is the global minimum for the function. Unlike a relative minimum, the absolute minimum is the lowest point on the entire graph of the function and cannot be surpassed anywhere else.
  • #1
harpazo
208
16
Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails.

f (x, y) = x^(2/3) + y^(2/3)

Solution:

f_x = 2/[3 (x)^1/3]

f_y = 2/[3 (y)^1/3]

f_xx = -2/[9 x^(4/3)]

f_yy = -2/[9 y^(4/3)]

f_xy = 0

I set f_x and f_y to 0 and found the critical point to be
(0, 0).

To find (0, 0, 0), I evaluated f (x, y) at the point (0, 0).

Can you please tell me what (0, 0, 0) represents here? I am confused about the critical point (0, 0) and the point in space (0, 0, 0). Are they the same point?

Is this ok so far?

d = -2/[9 x^(4/3)]*-2/[9 x^(4/3)] - [0]^2

I then evaluated d at the point (0, 0) and the result is 0.
This means the test fails.

The textbook goes on to say that there is absolute minimum in this case.

Is any of this correct? Why do we have absolute minimum here and not relative minimum?
 
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  • #2
The function:

\(\displaystyle f(x,y)=x^{\frac{2}{3}}+y^{\frac{2}{3}}\)

has no critical or stationary points. It has a "cusp" at $(x,y)=(0,0)$, which serves as the absolute or global minimum, and is otherwise unbounded (no global maximum).
 

1. What is a critical point of a multivariable function?

A critical point of a multivariable function is a point where the partial derivatives of the function are equal to zero. It is a point where the function could potentially have a maximum, minimum, or saddle point.

2. How do you find critical points of a multivariable function?

To find critical points of a multivariable function, you must first take the partial derivatives of the function with respect to each variable. Then, set each partial derivative equal to zero and solve the resulting system of equations to find the critical points.

3. What is an extremum of a multivariable function?

An extremum of a multivariable function is a point where the function has a maximum or minimum value. It can also refer to the maximum or minimum value itself.

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

To determine the nature of a critical point, you can use the second derivative test. You must first find the second partial derivatives of the function. Then, evaluate these second derivatives at the critical point. If the second derivatives are both positive, the critical point is a local minimum. If they are both negative, the critical point is a local maximum. If one is positive and one is negative, the critical point is a saddle point.

5. Can a multivariable function have multiple critical points?

Yes, a multivariable function can have multiple critical points. These points can be local maxima, local minima, or saddle points. It is important to consider all critical points when analyzing the behavior of a multivariable function.

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