Mathematics

Problem

Which of the following statements about a twice-differentiable function

f

is always true? a) If

f''(x)>0

for every

x

in an open interval, then the graph of

f

is concave up on that interval. b) If

f''(x)<0

for every

x

in an open interval, then

f

has a local maximum somewhere in that interval. c) If

f''(x)=0

for every

x

in an open interval, then the graph of

f

must contain a point of inflection in that interval. d) If

f'(x)>0

for every

x

in an open interval, then the graph of

f

is concave up on that interval.