Second Derivative Test for Partial Derivatives

In summary, the second derivative test for partial derivatives states that if the determinant D is positive and the second order derivative with respect to x is negative, then there is a local maximum at the critical point (a,b). If the determinant D is positive and the second order derivative with respect to x is positive, then there is a local minimum at the critical point (a,b). It does not matter which second order derivative is checked as long as they have the same sign in order for the determinant to be positive.
  • #1
SeannyBoi71
84
0
Hi there, just wanted to make a clarification before my final exam.

The second derivative test for partial derivatives (or at least part of it) states

if D = ∂2f/∂x2 * ∂2f/∂y2 - (∂2f/∂x∂y)2 and (a,b) is a critical point of f, then

a) if D(a,b) > 0 and ∂2f/∂x2 < 0, then there is a local max at (a,b)

b)if D(a,b) > 0 and ∂2f/∂x2 > 0, then there is a local min at (a,b)

and the other two parts are irrelevant for my question. My question is, do I have to specifically check that ∂2f/∂x2 is positive or negative, or can I check that ∂2f/∂y2 is positive or negative instead? i.e. does it really matter which one I check? Thank you in advance
 
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  • #2
Clearly, [itex]\left(\partial^2 f/\partial x\partial y\right)^2[/itex] is positive and you are subtracting it from [itex]\left(\partial^2f/\partial x^2\right)\left(\partial^2 f/\partial y^2\right)[/itex]. In order that the difference be positive, [itex]\left(\partial^2f/\partial x^2\right)\left(\partial^2 f/\partial y^2\right)[/itex] must be positive which means that [itex]\partial^2f/\partial x^2[/itex] and [itex]\partial^2 f/\partial y^2[/itex] must have the same sign.
 

Related to Second Derivative Test for Partial Derivatives

What is the Second Derivative Test for Partial Derivatives?

The Second Derivative Test for Partial Derivatives is a method used to determine whether a critical point of a multivariable function is a local minimum, local maximum, or saddle point. It involves taking the second partial derivatives of the function and evaluating them at the critical point.

How do you perform the Second Derivative Test for Partial Derivatives?

To perform the Second Derivative Test for Partial Derivatives, follow these steps:

  1. Find the critical points of the function by setting the first partial derivatives equal to zero.
  2. Calculate the second partial derivatives of the function.
  3. Substitute the critical point values into the second partial derivatives to obtain the Hessian matrix.
  4. Determine the eigenvalues of the Hessian matrix. If all eigenvalues are positive, the critical point is a local minimum. If all eigenvalues are negative, the critical point is a local maximum. If there are both positive and negative eigenvalues, the critical point is a saddle point.

What is the significance of the Second Derivative Test for Partial Derivatives?

The Second Derivative Test for Partial Derivatives is important because it allows us to classify critical points and determine the behavior of a multivariable function at these points. It helps us identify whether a critical point is a minimum, maximum, or saddle point, which can provide valuable information for optimization problems and understanding the behavior of a function.

Can the Second Derivative Test for Partial Derivatives be applied to all multivariable functions?

No, the Second Derivative Test for Partial Derivatives can only be applied to functions that are twice continuously differentiable. This means that the first and second partial derivatives of the function exist and are continuous at all points in the domain of the function.

Are there any limitations to the Second Derivative Test for Partial Derivatives?

Yes, the Second Derivative Test for Partial Derivatives may not always provide a definitive answer for the behavior of a function at a critical point. In some cases, the test may fail to classify a critical point because the eigenvalues of the Hessian matrix are zero or undefined. In these cases, further analysis may be needed to accurately determine the behavior of the function at the critical point.

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