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.
Technically not correct - the graph could have a "kink" like the function f(x)=|x| at x=0.
It's probably better to say that if the second derivative is negative, then the function is concave down. And also (though it's not needed for this problem) if the second derivative is positive, then the function is concave up.
- Hollywood