Below is the graph of a function y fx with several points l

Below is the graph of a function y = f(x) with several points labeled. Complete the chart by by determining at each point whether each of y, y\' and y\" is positive (+), negative (-), zero (0), or undefined (UNDEF). Which (if Any) of the points are local maxima? Which (if any) of the points are local minima? Which (if any) of the points are inflection points?

Solution

y is positive if it is above x axis and negative if below x axis

y\' is positive if y is increasing with increasing x and negative if y is decreasing with increasing x

y\'\' is positive if the graph is concave up and negative if graph is concave down and point of inflection if neither concave up or down

Point                                      y                                           y\'                                              y\'\'

A                                           -                                            +                                               +

B                                           +                                           UNDEF                                      UNDEF

C                                           -                                            0                                               +

D                                           +                                            +                                              0

E                                           +                                           0                                               -

F                                           +                                           -                                               -

b)

B and E

c)

A and C

d)

D