Let . The derivative is undefined at . Which statement about is most…

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

h(x)=x^{1/3}

. The derivative

h'(x)

is undefined at

x=0

. Which statement about

x=0

is most accurate? (a)

x=0

is a local maximum because

h'(0)

is undefined and

h''(x)<0

on both sides of

0

. (b)

x=0

is a local minimum because

h(0)=0

and

h(x)\ge 0

for all real

x

. (c)

x=0

is neither a local maximum nor a local minimum because

h(x)

increases on both sides of

0

. (d)

x=0

is the global maximum of

h

because the function is unbounded below as

x\to -\infty