<p>Over a closed interval, when do you consider endpoints in your final answer. For example, when they ask for relative extrema, or upward concavity; when is an endpoint considered for those things?</p>
<p>I used to ALWAYS consider endpoints, but this year's response (question 4) REALLY confused me. They considered the endpoint for increasing/decreasing, but not for concavity. Can you list the items for which to consider endpoints?</p>
<p>Questions :
<a href="http://apcentral.collegeboard.com/apc/public/repository/b_calculus_bc_frq_02_10318.pdf%5B/url%5D">http://apcentral.collegeboard.com/apc/public/repository/b_calculus_bc_frq_02_10318.pdf</a></p>
<p>Answers:
<a href="http://apcentral.collegeboard.com/apc/public/repository/b_sg_calculus_bc_02_11408.pdf%5B/url%5D">http://apcentral.collegeboard.com/apc/public/repository/b_sg_calculus_bc_02_11408.pdf</a></p>
<p>I’m not sure, but I think it’s because you don’t know if f’(x) is differentiable at the end points.</p>
<p>I’m not sure if I can list when to do endpoints comprehensively, but I am pretty sure you do for absolute/global maximums/minimums definitely.</p>
<p>I think you don’t check for local mins/maxs and for concavity.</p>
<p>A lot of the confusion that occurs is in large part due to the differences in definitions across different texts and sources.</p>
<p>With global/absolute maxes and mins, you defininitely have to check the endpoints, as those are one place where those could occur.</p>
<p>With increasing/decreasing intervals, the answers will usually occur with endpoints included. That being said, omission of endpoints won’t harm your score. The reason that endpoints are included is because, technically, a function is increasing whenever x1 < x2, then y1 < y2. For instance, the function y = x^3 is increasing at x = 0, because there’s no interval <a href=“with%20x%20%3C%200”>x, 0</a> or <a href=“with%20x%20%3E%200”>0, x</a> where f(x) doesn’t meet the criteria above. This happens despite the fact that y’(0) = 0.</p>
<p>Concavity is usually defined based on the sign of y", and accordingly, endpoints aren’t included (since places where y" = 0 is neither positive nor negative). I’ve seen the definition of concavity where a function is concave up when f ’ is increasing and concave down when f ’ is decreasing. According to that definition, endpoints would be included, though I honestly don’t know if that would be scored that way.</p>
<p>As for relative maxima/minima, some texts allow endpoints to be considered and others don’t. Frequently, relative maxima/minima questions are asked on open intervals so that endpoints are irrelevant.</p>
<p>Hope that helps.</p>
<p>MathProf, you are a lifesaver. I had interpreted my question to be a matter of definition, but after searching online and through my textbook nothing gave me a clear or definate answer. This helps a lot; I’ll remember it for the test. Thanks!</p>
<p>when they ask for an absolute one. relative min/max, you can ignore them.</p>