Suneil P. answered • 06/19/14
Knowledgeable and Passionate University of Pennsylvania Math Tutor
Rather than provide you with the exact answer, let us first review some important points and see how we can proceed from there:
Recall the quotient rule (which can also be understood as a consequence of chain and product rules): where f and g are two functions of x, g not equal to 0, (f/g)'=(gf'-fg')/g^2.
To analyze concavity, we should examine the second derivative of the function (the rate of change of the rate of change); this makes sense, since a "concave up" function has an increasing rate of change (or positive second derivative) while a "concave down" one has a decreasing rate of change (or negative second derivative). So sign of second derivative reasonably provides a test for concavity.
Inflection points are points at which concavity instantaneously changes from + to - (or vice-versa). So, we would expect the second derivative at such a point to be 0 (or possibly undefined). However, a 0 second derivative is not sufficient to determine whether that point is an inflection point (we could have a case in which the second derivative goes from + to 0 to + again without hitting the negatives, like the function x^2). What we can do however for sufficiency, is look at all candidates: points with 0 or undefined second derivative and then look at points to the left and right of these candidates; if the sign of the second derivative changes when crossing the candidate, the candidate is an inflection point.
To summarize, take the second derivative; at this point, it may be helpful to make a number line and plot points with 0 or undefined second derivative and then make note of the sign of the second derivative for each interval between two candidate points. Then all three qs can easily be answered.
Hope this helps.
