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## Multivariable calculus

### Course: Multivariable calculus>Unit 3

Lesson 3: Optimizing multivariable functions

Just because the tangent plane to a multivariable function is flat, it doesn't mean that point is a local minimum or a local maximum. There is a third possibility, new to multivariable calculus, called a "saddle point". Created by Grant Sanderson.

## Want to join the conversation?

• You said this is new for multivariable calculus - but isn't there something called inflection point in one dimension that is the equivalent of a saddle point? The tangent is flat but it's neither a maxima nor a minima. • That is a very insightful observation. However, you need some clarifications.

First, not all inflection points are "saddle points." Only inflection points that are flat, where the slope is zero, can be confused with relative extrema. That's why you also have to check whether the slope's sign changes or the function's concavity when deciding if you have an extremum.

Second, you can have "flat inflection points" in multivariable calculus too. For example, f(x,y) = x^3 + y^2 at (0,0). They are not saddle points, because the problem is not a disagreement between the inputs, but just a problem with one of the inputs (in this case, x). (Though of course there may be problems with each of the inputs, as in f(x,y) = x^3 + y^3.)

You can't really have saddle points in one dimension (of input). "Flat inflection points" are called something else and for good reason (though I don't know what they're usually called other than "flat inflection points" and the super annoying thing in calculus I have to check every time and I hate so much). However, they both show how cautious you need to be when finding extrema of functions (which is why I hate them so much).

UPDATE:
Okay, I looked on Wikipedia and apparently "flat inflection points" are saddle points too (even in single var!). It's just their definition: they call any point where the gradient is zero but is not a local extremum a saddle point, even if they do not resemble saddles. https://en.wikipedia.org/wiki/Saddle_point
; p • at you said local minima - think you meant local maxima. Not a big deal, just in case anyone gets confused. • For single variable, there is a saddle point as well.
Consider F(x) = x^3
df/dx = 3x^2 equating by zero then we have an extremum point at x=0
getting the second derivative at this point we found it equal to zero, which is neither max nor min point also from the graph it is clear that this point is a saddle point. • Why do I keep thinking of Inflection Point... (1 vote) • In single-variable calculus world, would a function that has a derivative evaluated at a point equal to 0 but a second derivative evaluated at a point equal to 0 and either increasing or decreasing (like x^3 evaluated at 0) be considered a saddle point? It is also neither a local minimum or maximum, so what would we consider it as? Thanks!
(1 vote) • It's interesting to consider second directional derivatives at saddle-points -- there are interesting conditions for what direction causes the point be be a minima and what direction causes it to be a maxima.
(1 vote) • An idea: in case of the one variable functions, it could also be an option for the tangent to be horizontal if the function changes its band. Like the x^3. In x=0 point its derivate is actually zero without being a local maximum or minimum. (time )

So a saddle point is named after its shape, but if we take the x^3 and y^3 az the surface of the function (I don't know what the expression could be f(x;y) = ?) then taking the formal definition, their derivates f'_x and f'_y would be zero at the point (0;0), it would not be a local max or min, and it wouldn't have the shape of a saddle... What's then? Does it have a special name? :-D
(1 vote) • If the partial derivatives are all zeros, is the gradient of this function pointing nowhere?
(1 vote) 