Derivatives and Second Derivatives

A function f satisfies the following conditions...
f(4) = 28
f '(4) = 3
f ''(x) < 0 for x >or= 4

Which of the following are possible values for f(6)? (Select all that apply.)
a. 31
b. 34
c. 37
d. None of these

Quote:

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Originally Posted by melliep

f ''(x) < 0 for x 4

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For $\displaystyle x\;\;?\;\;4$? Something is missing here.

And if someone does reply, could you please tell me why you got the answer you did?

Thank you.

Unless you are missing some more information, they all seem possible (except of course d). The local behavior of a function over $\displaystyle \mathbb{R}$ does not determine its global behavior (this is different for a holomorphic complex functions).

So the answer is that $\displaystyle f(6)$ could be anything.

Good point but not quite correct. We are given some "global" information- we are told that f"(x) < 0 for all x> 4. Therefore f' is less than 3 for x> 4 and so the graph of f(x) is below the tangent line at x= 4. That tangent line is y= 3(x- 4)+ 28 and when x= 6 that gives 6+ 28= 34. f(6) must be less than 34.

Quote:

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Originally Posted by HallsofIvy

Good point but not quite correct. We are given some "global" information- we are told that f"(x) < 0 for all x> 4.

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Yes, thank you of course you are right. I seem to have mis-read the post.