If the situation is reversed and the sign of y′ changes from − to + as x
increases through c, then c yields a local minimum. Figure N4–3b shows
this case. The schematic sign pattern of y′, + 0 − or − 0 +, describes each
situation completely. If y′ does not change sign as x increases through c,
then c yields neither a local maximum nor a local minimum. Two examples
of this appear in Figures N4–3c and N4–3d.Figure N4–3(5) Find all x’s for which y′′ = 0; these are x-values of possible points of
inflection. If c is such an x and the sign of y′′ changes (from + to − or from −
to +) as x increases through c, then c is the x-coordinate of a point of
inflection. If the signs do not change, then c does not yield a point of
inflection.
The crucial points found as indicated in (1) through (5) above should be
plotted along with the intercepts. Care should be exercised to ensure that the
tangent to the curve is horizontal whenever = 0 and that the curve has the