Finding Local Extremes Of A Function; Using The Derivative - HP -15C Advanced Functions Handbook

Section 1: Using ("SOLVE | Effectively
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You can then use | SOLVE |to find the root x of this equation (for any
given value of n, the number of the root). Knowing x, you can
calculate the corresponding value of y.
A final consideration for this example is to choose the initial
estimates that would be appropriate. Because the argument of the
inverse cosine must be between -1 and 1, x must be more negative
than about -0.1059 (found by trial and error or by using | SOLVE |).
The initial guesses might be near but more negative than this
value, -0.11 and -0.2 for example.
(The complex equation used in this example is solved using an
iterative procedure in the example on page 81. Another method for
solving a system of nonlinear equations is described on page 122.)
Finding Local Extremes of a Function
Using the Derivative
The traditional way to find local maximums and minimums of a
function's graph uses the derivative of the function. The derivative
is a function that describes the slope of the graph. Values of x at
which the derivative is zero represent potential local extremes of
the function. (Although less common for well-behaved functions,
values of x where the derivative is infinite or undefined are also
possible extremes.) If you can express the derivative of a function
in closed form, you can use | SOLVE | to find where the derivative is
zero—showing where the function may be maximum or minimum.
Example: For the design of a vertical broadcasting tower, radio
engineer Ann Tenor wants to find the angle from the tower at
which the relative field intensity is most negative. The relative
intensity created by the tower is given by
cos(2?r/icos d) — cos(2?r/i)
E = -
-cos(27r/i)]sin0
where E is the relative field intensity, h is the antenna height in
wavelengths, and 6 is the angle from vertical in radians. The
height is 0.6 wavelengths for her design.
The desired angle is one at which the derivative of the intensity
with respect to 6 is zero.  