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Problem : Find the second derivative of f (x) = x3.

We begin by computing the first derivative:


f'(x)=  
 =  
 =3x2  

The derivative of this new function will be the second derivative of f.


f''(x)=  
 =  
 =6x  

Problem : For which value of x is f''(x) = 0, if f has the following graph?

Figure %: Locate the Points for which f''(x) = 0

The slope of the graph is decreasing for x < 2, so in this region f''(x) < 0. Similarly, the slope of the graph is increasing for x > 2, so f''(x) > 0 here. Thus at x = 2, we must have f''(x) = 0 (by the intermediate value theorem).

Problem : Find all functions f (x) such that f''(x) = 0.

The second derivative being zero means that the slope of the graph of the function is constant for all x. The only graphs for which this is true are the lines, so f (x) = mx + b for some constants m and b.
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