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The Derivative

Problems

The Second Derivative

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Problem : Find the second derivative of f (x) = x 3 .

We begin by computing the first derivative:


f'(x) =  
  =  
  = 3x 2  

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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