Problem+30+Elie+and+James+B

Suppose f(3)=2, f'(3)=½, and f'(x)>0 and f"(x)<0 for all x.

A) Sketch a possible graph for f.



B) How many solutions does the equation f(x)=0 have? Why?

1. Because the slope never changes from positive to negative, the curve will always be increasing and never cross the x axis more than one time.

C) Is it possible that f'(2)=1/3? Why?

No. Since the second derivative is negative and the first derivative is positive, the first derivative is decreasing. The slope of the function, in fact, begins as a very large positive number and approaches zero. And f' (3) is given to be ½ so f'(2) can not be 1/3 because 1/3 is less than ½.