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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).
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The
derivative
of f(x) at the point x is equal to the slope of
the tangent
to y = f(x) at x.
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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.
These basic properties of the maximum and minimum are summarized in the
following table.
| Behaviour |
Graphs |
Derivative
(slope of tangent)
at point slightly to the
left of the maximum
point x0 |
Derivative
(slope of tangent)
at maximum point x0 |
Derivative
(slope of tangent)
at point slightly to the
right of the maximum
point x0 |
| Local maximum |
 |
f ´(x0−)
> 0
(positive, increasing)
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f ´(x)
= 0
(zero)
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f ´(x0+)
< 0
(negative, decreasing)
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| Local minimum |
 |
f ´(x0−)
< 0
(negative, decreasing)
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f ´(x)
= 0
(zero) |
f ´(x0+)
> 0
(positive, increasing)
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Exercise
Test that the properties stated in the above table are true. You can examine
the examples provided in the scroll bar on the top of the applet below or enter
your own function in the box provided. If you enter your own function, you must
use the symbols + for add, - for subtract, * for multiply, / for divide,
and ^ to raise to a power. You can also use various mathematical functions: sin,
cos, tan, sec, cot, csc, arcsin, arccos, arctan, exp, ln, log2, log10, abs, sqrt
and cubert. (Here, "abs" is the absolute value function, "sqrt"
is the square root function and "cubert" is the cube root function.)
Make sure you understand the following connections
between the two graphs.
- When the graph of the function f(x) has a horizontal tangent then
the graph of its derivative f '(x) passes through the x
axis (is equal to zero).
If the function goes from increasing to decreasing, then that point is a local
maximum.
If the function goes from decreasing to increasing, then that point is a local
minimum.
Also, as we learned previously
- 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).
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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.
- 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.
Exercise
To see some worked examples, get a new exercise and immediately click show
answer until you are confident.
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