When sketching the graph of a function, the obvious first step is to plot a few points. Specifically, the zeros, the turning points, and the inflection points (and, for good measure, the y-intercept). But how does one connect the dots? The answer lies within three quantities: function value, slope, and concavity. In fact, it’s just their signs that matter.
The values of `f(x)`, `f'(x)`, and `fʺ(x)` represent function value, slope, and concavity, respectively. This table summarizes the meaning of their signs:
|Value||`= 0`||`> 0`||`< 0`|
|`fʺ(x)`||inflection point||concave up||concave down|
Here is an example of the information that these three quantities, given by the function and its first two derivatives, provides you with: