sketch the graph of a function f having given characteristics
f(2) = f(4) = 0
f(3) is defined.
f'(x) < 0 if x < 3
f'(3) does not exist.
f'(x) > 0 if x > 3
f"(x) < 0, x does not equal 3
If the derivative doesn't exist at a point it means there is a discontinuity.
If you have a positive derivative it means the function is increasing: if it is negative then it is decreasing.
If f(3) is defined but f'(3) doesn't exist, then it means you have either a discontinuity in the graph or the graph itself has a "kink" in it and isn't smooth (but is still continuous).
If second derivative is increasing then first derivative is increasing: if decreasing then derivative is decreasing.
There are many solutions to this problem graphically and function-wise but they will have the attributes outlined with the above characteristics of derivatives.