Why 2nd derivative can find concavity

[vanmaiden]

Homework Statement

I know one can find the critical points by taking whatever points make the first derivative zero or undefined. I also know that when you put the critical points into the second derivative the positivity or negativity of the number will tell you whether the graph will be concave up or down. What gets me is that the critical point you put into the 2nd derivative has a slope of zero. How can you find out the concavity from a point with a slope of zero?

Homework Equations

derivative properties, addition rule

The Attempt at a Solution

I have been thinking about this for awhile and it stumps me on why this way to find the concavity actually works.

 

[HallsofIvy]

You can't! That's not true. For example, the function $f(x)= x^2$ is "concave upward" for all x but the second derivative is 2, not 0, for all x. A function is "concave upward" as long as the second derivative is positive, concave downward if the second derivative is negative. The second derivative is 0 where the concavity changes.

 

[vanmaiden]

A function is "concave upward" as long as the second derivative is positive, concave downward if the second derivative is negative.

i see, but what makes the second derivative have this property?

 

[HallsofIvy]

The second derivative is, of course, the rate of change of the first derivative. If the second derivative is positive then the first derivative is increasing- it starts low on the left so you have almost a horizontal line then increases so it becomes steeper and steeper. That gives you a "concave upward" graph. If the second derivative is negative, it is just the opposite- the graph starts upward sharply, then starts to level off- that is "concave downward".

 

[vanmaiden]

So say I was to put a random x value into the second derivative and it spat out 24. Then, I put in a different random x value and it spat out say...26. What would the difference be between the two points?