The Sign of the Derivative

Recall from the previous page: Let f (x ) be a function and assume that for each value of x. we can calculate the slope of the tangent to the graph y = f (x ) at x. This slope depends on the value of x that we choose, and so is itself a function. We call this function the derivative of f (x ) and denote it by f ´ (x ).

Maximum and Minimum

The graph of a function y = f (x ) has a local maximum at the point where the graph changes from increasing to decreasing. At this point the tangent has zero slope. The graph has a local minimum at the point where the graph changes from decreasing to increasing. Again, at this point the tangent has zero slope .

Exercise

Make sure you understand the following connections between the two graphs.
  • When the graph of the function f (x ) has a horizontal tangent then
  • When the gradient of the function f (x ) is positive ,
the graph of its derivative f '(x ) is above the

x -axis (is positive) .

  • When the gradient of the function f (x ) is negative.

    the graph of its derivative f '(x ) is below the x -axis (is negative) .

  • Can't see the above java applet? Click here to see how to enable Java on your web browser. (This applet is based on free Java applets from JavaMath )

    This gives a method for finding the minimum or maximum points for a function. See later for the preferred method .

    1. Differentiate the function, f (x ). to obtain f '(x ) .
    2. Solve the equation f '(x ) = 0 for x to get the values of x at minima or maxima.
    3. For each x value:
      1. Determine the value of f '(x ) for values a little smaller and a little larger than the x value.
      2. Decide whether you have a minimum or a maximum.
      3. Calculate the value of the function at the x value.

    Exercise

    To see some worked examples, get a new exercise and immediately click show answer until you are confident.

    Source: mathsfirst.massey.ac.nz

    Category: Bank

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