Second partial derivative test intuition

TL;DR

The second partial derivative test helps determine if a critical point of a multivariable function is a local maximum, local minimum, or a saddle point.

Transcript

• [Voiceover] Hey everyone. So in the last video I introduced this thing called the second partial derivative test, and if you have some kind of multivariable function or really just a two variable function is what this applies to, something that's f of x, y and it outputs a number. When you're looking for places where it has a local maximum or a l... Read More

Key Insights

• 😥 The second partial derivative test helps determine the nature of a critical point without graphing the function.
• 😀 It involves computing a value "H" using second partial derivatives and the mixed partial derivative.
• 🥺 disagreement in concavity between the x and y directions leads to a saddle point.
• 🫤 The mixed partial derivative captures the diagonal disagreement in the function's behavior.
• ❣️ The test provides a way to determine whether the agreement or disagreement between the x and y directions dominates in determining the type of the critical point.
• 🍉 Only three different terms are needed to account for disagreement in infinitely many directions.
• 🏆 The second partial derivative test can be justified rigorously, and more details can be found in an article.

Questions & Answers

Q: What are critical points in relation to the second partial derivative test?

Critical points are where the gradient of a function equals zero, and they can be considered as potential local maxima, minima, or saddle points.

Q: How is the value "H" computed in the second partial derivative test?

The value "H" is computed by evaluating the second partial derivatives of x and y, as well as the mixed partial derivative, at the critical point. It is calculated as the product of the pure second partial derivatives minus the square of the mixed partial derivative.

Q: What does it mean if the value of "H" is greater than zero?

If the value of "H" is greater than zero, it indicates that the critical point is either a local maximum or a local minimum. The concavity in a specific direction, determined by the sign of the second partial derivative with respect to that direction, determines the type.

Q: What does it mean if the value of "H" is less than zero?

If the value of "H" is less than zero, it signifies that the critical point is a saddle point. A saddle point is neither a local maximum nor a local minimum and represents disagreement in different directions.

Summary & Key Takeaways

• The first step in finding local maxima or minima is to find critical points where the gradient equals zero.

• The second partial derivative test involves computing a value called "H" using second partial derivatives and the mixed partial derivative.

• If H is greater than zero, the point is either a local maximum or a local minimum, and the concavity in one direction determines the type. If H is less than zero, it is a saddle point.