Mathematics

Problem

Consider the function

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

which is defined for all real

x

. (a) Show that

h

is continuous at

x=0

. (b) For

x\neq 0

, compute

h'(x)

. Present your work separately for

x>0

and

x<0

if necessary. (c) Evaluate

\displaystyle\lim_{x\to 0^-}h'(x)

and

\displaystyle\lim_{x\to 0^+}h'(x)

. (d) Does

h

have a tangent line at

x=0

? If not, explain which term of

h

is responsible for the failure of differentiability and characterize the type of nondifferentiable point observed.