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 .
ExerciseMake 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 ,
x -axis (is positive) .
the graph of its derivative f '(x ) is below the x -axis (is negative) .
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This gives a method for finding the minimum or maximum points for a function. See later for the preferred method .
- Differentiate the function, f (x ). to obtain f '(x ) .
- Solve the equation f '(x ) = 0 for x to get the values of x at minima or maxima.
- For each x value:
- Determine the value of f '(x ) for values a little smaller and a little larger than the x value.
- Decide whether you have a minimum or a maximum.
- Calculate the value of the function at the x value.
To see some worked examples, get a new exercise and immediately click show answer until you are confident.Source: mathsfirst.massey.ac.nz