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### Course: Multivariable calculus>Unit 3

Lesson 3: Optimizing multivariable functions

# Warm up to the second partial derivative test

An example of looking for local minima in a multivariable function by finding where tangent planes are flat, along with some of the intuitions that will underly the second partial derivative test. Created by Grant Sanderson.

## Want to join the conversation?

• Can I use laplacian to find a multivariable maxima and minima? Because in the videos about the laplacian, it says that laplacian can determine whether that point is lower or higher than other points surrounding it?
• I think we can't. Because laplacian just evaluate the value of the area around the point which is analogous to the average value.
• At , we conclude that neither the second partial derivative with respect to x, nor the first partial derivative with respect to x are a function of y. So, does that mean that the steepness and concavity of the graph-plane intersection do not depend on what y-value we slice at? But doesn't the curve look much steeper, for example, when we slice at y = sqrt(2) than when we slice at y = 0?
• Yes, neither the second partial derivative with respect to x nor the first partial derivative with respect to x are dependent on y. But remember, the function of interest is dependent on both *x* and y. Thus, in order to truly understand the steepness and concavity of the entire 3d function, we must also examine the first and second partial derivatives with respect to y. I hope this cleared things up :)
If it didn't, please let me know, as this is an extremely important conceptual issue you raised and an understanding of this concept is crucial to an understanding of multi-variable calculus.
• So are there extreme points that would technically not count as a max, min, or a saddle-point? And if so, what do they look like?
• Yes, such a point can be constructed.

Here's a 1 degree-of-freedom example:
consider f(x)=sin(x)-x.
At the origin the first and second derivatives are both zero, but it is not a maximum nor a minimum.

Moving up one degree-of-freedom, consider the function:
f(x,y)=sin(x)-x+sin(y)-y.
Again, at the origin all first and second derivatives are zero, but it is not a minimum, maximum, or saddle.
I wish these comments would let me post a graph for you, as it's quite easy to see that the 1st and 2nd derivatives are zero at the origin, but that it is not a maximum nor a minimum nor a saddle. It is an inflection point.
• Do we strictly need the second derivative to determine if we have a minima or maxima? Why can't we just enter the solution into the original equation to find the value of the equation at those points and just compare these values to see which is the maxima, minima or saddle points?

For example the solution for $f(x,y) = x^4-4x^2+y^2$ is: ${(0, 0), (0, \sqrt{2}), (0, -\sqrt{2})}$

And...

$f(0, 0) = 0$
$f(0, \sqrt{2}) = 4-4*2+0 = -4$
$f(0, -\sqrt{2}) = 4-4*2+0 = -4$

Therefore: (0, 0) is a local maxima or saddle point
While the other points are local minima.
• You can, but similar to the second derivative test from single-var calc, you can use the test to determine is a specific point is special, i.e. a saddle point, etc. Which you can't do by simply plugging in the numbers.