Practice Ab Calc Questions
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What is local extrema in Calculus (AP Calc AB question)?
one of my FRQ practice questions asks if the given f(x) “has a local maximum, local minimum, or neither at this point”. I looked at the answer, but it doesn’t explain what local extrema is. Non-wikipedia answer please 🙂 thanks!
The link to the actual question is below (it’s #5)
http://apcentral.collegeboard.com/apc/public/repository/b_sg_calculus_ab_02_11407.pdf
A local extremum is either a local maximum or a local minimum. Definitions:
The point (x0, f(x0)) is a local maximum of f(x) if there is some neighborhood of x0 such that f(x0) is the unique maximum value of f in that neighborhood.
The point (x0, f(x0)) is a local minimum of f(x) if there is some neighborhood of x0 such that f(x0) is the unique minimum value of f in that neighborhood.
When I write “unique maximum value” I mean that the maximum value is attained only at the single x-value x0, and nowhere else in the neighborhood. Same for “minimum”.
It is the “some neighborhood of x0” part that makes the maximum or minimum local, because there might be values of f(x) greater than the local maximum, or less than the local minimum, somewhere outside that neighborhood.
By Fermat’s theorem, any local extremum of a differentiable function defined on an open interval must occur at a point where the derivative of the function is zero.
AP / AB Calculus Test – Sample Questions 23 & 24
