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Suppose that the second derivative of a function is continuous, is positive on x <2, is negative on x > 2, and is zero at x = 2. What can you say about the behavior of the function at x = 2?

A. f is a maximum at x = 2

B. f is a minimum at x = 2

C. f changes inflection at x = 2

D. There is not enough information given.

I think the answer for this is C but im not really sure. If the second derivative changes between 2 ( before 2 and after) then its changing from concave up to concave down which means there is an inflection change. Is C the right answer?

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Looks right, but maybe for completeness come up with examples satisfying hypothesis but where the conclusions of A,B are false. As for D, ruling it out means just knowing definition of inflection. – coffeemath Jul 16 '14 at 0:12
up vote 1 down vote accepted

We know what you said about the second derivative. So what does this imply for the first one? Well before $x=2$, the function must be rising! why, because its derivative is positive up to $x=2$ (the 2nd derivative is the first derivative of the first derivative). At $x=2$, it must be stationary. After $x=2$, it must be decreasing as its derivative is negative.

So now this tells that the first derivative has a local maximum at $x=2$, but it doesn't tell us if the maximum is at a positive or a negative (y is greater or lesser than 0). It could be at any arbitrary y value!

So for the actual function,it might not be flat at all at $x=2$ !!! Instead we know for sure that the slope after $x=2$ starts "curling" to the opposite direction from the direction it was curing from, and note, it may have never even become tangential to the x-axis (horizontal)! So if the function's line was the trace of a boat, before $x=2$ it may have been turning left and after $x=2$ it must have started turning right - that's all we can know!

So to your question: clearly the only thing we can deduce from the given information is that there is an inflection at $x=2$

Have a nice day :)

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Yes, C is correct. By definition, an inflection point is one where the concavity (second derivative) changes. (In all this I assumed that you mis-copied C, which probably said "f has an inflexion point at 2.")

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