Second derivative test | Using derivatives to analyze functions | AP Calculus AB | Khan Academy

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Sal justifies the second derivative test, which is a way of determining relative minima & maxima, and gives an example.

AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.

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This makes sense if you’re looking at how the tangents change from left to right on the curves. If you look at the tangents from right to left it breaks down. On the concave down shape, from right to left, the tangent is decreasing to 0 then getting steeper but in a positive way.

ethandiamond

Actually, there are TWO cases where the Second Derivative Test is inconclusive: (1) f"(c) = 0 (you have that one), and (2) f"(c) undefined (you missed that one). It IS possible for f'(c) to be DEFINED but f"(c) to be UNDEFINED. For example, consider the function f(x) = x^(4/3). Then f'(x) is essentially x^(1/3) which is 0 only for x = 0, so x = 0 is a critical number (the only one, since f'(x) is never undefined). Then, f"(x) is essentially x^(-2/3), which is UNDEFINED for x = 0, so the Second Derivative Test is INCONCLUSIVE for the only critical number for f, x = 0. However, if we use the First Derivative Test, we find that f'(x) is negative for x to the left of 0 and positive for x to the right of 0, so clearly (0, 0) is a local MINIMUM point.

trwent

Bless Khan Academy during corona virus pandemic 🙏🙏🙏

mattwarrenAV

Why is 2nd derivative of functions -ve for max and +ve for min

mohsinchaudhary

What is the higher operation needed to determine the concavity of a function with a critical point of 0?

freyawoodrow

this might be obvious but how did you get -4 for the second derivative?

nextgencs

wait. Isn't the second diff of a quadratic a point?

nohackers

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