<p>1) The function f is twice differentiable, and the graph of f has no points of inflection. If f(6) = 3, f '(6) = -1/2, and f ''(6) = -2, which of the following could be the value of f(7)?</p>
<p>A) 2 B) 2.5 C) 2.9 D) 3 E) 4</p>
<p>A
10 charactersss</p>
<p>Can you please explain? I am having a hard time…</p>
<p>Yeah I would appreciate it if you explained also, I got B but I’m pretty sure that that’s the wrong answer…just a gut feelin’.</p>
<p>Yeah A is correct. I have no idea why though. Could someone please explain? :)</p>
<p>A. since f’'(6) is negative, and there are no points of inflection, the second derivative has to continue to be negative. Thus the average rate of change has to be even less than -.5, and the only choice less than 2.5 is 2.</p>
<p>I’ll explain. f’‘(x) (assume the variable is x to prevent complication) is negative here. And since there’s no point of inflection, it’s concave down, so f’(x) keeps decreasing. Notice that the slope is -1/2. Since the slope keeps decreasing, the change in y as x changes will be greater than -1/2, so the only viable choice was A.</p>
<p>I’m terrible at explaining stuff, but I’ll try to clarify more if you don’t get it.</p>
<p>Aww wow I feel silly. Thanks Chanto.</p>
<p>We know that f(6) = 3, and we know that the first derivative at 6 is -1/2. This tells us that at f(7) the function will have decreased by 1/2, assuming its linear. So f(7) would be 2.5 had the function been linear, but we also know that the second derivative is -2. This tells us that the value of f(7) must be less than 2.5, and the only value remaining is 2.</p>
<p>no problem. these word problems are the hardest type, haha</p>
<p>I got A as well.</p>
<p>The derivate (slope) of the function at 6 is negative and the derivative of the derivative is negative too, meaning that the function will continue decreasing after x = 6.</p>
<p>This should eliminate D and E as they have y values larger than those at x = 6. </p>
<p>The problem gave you the point (6, 3) and asked for (7, y)</p>
<p>Finding the slope of those two points, you get (y-3) / (7 - 6) = y - 3 </p>
<p>y - 3 < - (1/2) because the second derivate is negative</p>
<p>y < 2.5</p>
<p>2 is the only answer that fits.</p>
<p>Sorry if that isn’t how you’re supposed to do this problem…it’s just how I did it.</p>
<p>Ouch, looks like I have to review what the second derivative tells you more lol, thanks Chanto and pdawggy!</p>