Using Repeated Estimation - HP -15C Advanced Functions Handbook

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Section 1: Using [SOLVE] Effectively
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Using Repeated Estimation
A third technique is useful when it isn't practical to calculate the
derivative. It is a slower method because it requires the repeated
use of the [SOLVE] key. On the other hand, you don't have to find a
good value for A of the previous method. To find a local extreme of
the function f(x), define a new function
g(x) = f(x)-e
where e is a number slightly beyond the estimated extreme value of
f(x). If e is properly chosen, g ( x ) will approach zero near the
extreme of f(x) but will not equal zero. Use | SOLVE | to analyze g ( x )
near the extreme. The desired result is Error 8.
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If Error 8 is displayed, the number in the X-register is an x
value near the extreme. The number in the Z-register tells
roughly how far e is from the extreme value of f(x). Revise e to
bring it closer (but not equal) to the extreme value. Then use
[SOLVE | to examine the revised g(x) near the x value previously
found. Repeat this procedure until successive x values do not
differ significantly.
If a root ofg(x) is found, either the number e is not beyond the
extreme value of f(x) or else [SOLVE| has found a different
region where f(x) equals e. Revise e so that it is close to—but
beyond—the extreme value of f(x) and try [SOLVE | again. It
may also be possible to modify g ( x ) in order to eliminate the
distant root.
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