(a) y ′(c) = 0; y ′′(c) > 0;
c yields a local minimum.(b) y ′(c) = 0; y ′′(c) = 0;
c yields a local minimum.
FIGURE N4–2
(4) If y ′(c) = 0 and y ′′(c) = 0, investigate the signs of y ′ as x increases through c. If y ′(x) > 0 for
x’s (just) less than c but y ′(x) < 0 for x’s (just) greater than c, then the situation is that indicated
in Figure N4–3a, where the tangent lines have been sketched as x increases through c; here c
yields a local maximum. 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.