Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

92 Extrema and the Geometric Applications of a Derivative [Ch. 3
For the abscissa of the point of inflection x of the graph of a function
y f (x) there is no second derivative f (* = ) or /" (x ). Points at which
f'(x) Q or f (x) does not exist are called critical points of the second kind.
The critical point of the second kind x is the abscissa of the point of inflection
if I" (x) retains constant signs in the intervals x 6 < * < * an
.d
x < x < Jc
? + 6, where 6 is some positive
number; provided these signs are
y~f(x)
I
bx a,
Fig. 29
I
i
b,
X
opposite. And it is not a point of
inflection if the signs of f (x) are the
same in the above-indicated intervals.
Example 1. Determine the intervals
of concavity and convexity and
also the points
Gaussian curve
of inflection of the
and
Solution. We have
Equating the second derivative y* to zero, we find the critical points of tHe
second kind
* = 7=r and *o = T=-
These points divide the number scale OO