Mathematics

Problem

Consider

h(x)=\sqrt[3]{x}+|x|

. Which statement correctly describes the behavior of

h

x=0

? a)

h

is continuous at

x=0

, but

h'(x)

approaches

+\infty

from the right and

-\infty

from the left, so the graph has a cusp at the origin. b)

h

is continuous at

x=0

and has a corner there because the

|x|

term yields slopes

+1

and

-1

. c)

h

is continuous at

x=0

, but not differentiable; both one-sided derivatives grow without bound to

+\infty

, giving a vertical tangent at the origin. d)

h

is not continuous at

x=0

because

|x|

is not differentiable there.