Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of c at which the basic shape of the curve changes. f ( x) = In(x2+ c) We see that if c c(-1 ) 1x/ > - C , and lim f(x) = - 00 lim f ( x ) = - OO , since In(y) - -was y - 0. x- V-ct x->-V-c Thus, for c 0, there are no asymptotes. To find the extrema and inflection points, we differentiate, as follows. 2x f ( x ) = In(x2 + c ) = f' ( x ) = + C So by the First Derivative Test, there is a local and absolute minimum at X = 0. 2(-x2 + ex + c ) Differentiating again, we get f"(x) = ( x 2 + c ) 2 X Now if c 0, then f" changes sign when c = 4 X = C 1 + Your answer cannot be understood or graded. More Information (entered as a comma-separated list). 2x X X So for c > 0 there are inflection points at x = (No Response) X (entered as a comma-separated list), and as c increases, the inflection points get further apart