What is a saddle point?

Answer 1

Coming from one direction it looks like we've hit a maximum, but from another direction is looks like we've hit a minimum.

Here are 3 graphs:

#y = x^4# has a minimum at #x=0#

graph{y = x^4 [-12.35, 12.96, -6.58, 6.08]}

#y = -x^2# has a maximum at #x=0#

graph{-x^2 [-12.35, 12.96, -6.58, 6.08]}

#y = x^3# has a saddle point at #x=0#

graph{x^3 [-12.35, 12.96, -6.58, 6.08]}

Coming from the left it looks like a maximum, but coming from the right it looks like a minimum.

Here's one more for comparison:

#y=-x^5#

graph{-x^5 [-10.94, 11.56, -5.335, 5.92]}

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Answer 2

A saddle point is a critical point of a function where the gradient is zero but it's not a local minimum or maximum. In other words, it's a point where the function stops increasing or decreasing along one direction but continues along another. At a saddle point, the function resembles a saddle shape in the vicinity of that point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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