Mathematics

Problem

Let

h(x)=\begin{cases} kx^{2}, & x<0,\\ kx, & x\ge 0, \end{cases}

where

k\neq 0

is a constant. Which statement about the differentiability of

h

x=0

is correct? a)

h

is differentiable at

x=0

for every nonzero

k

because both one-sided derivatives are

0

. b)

h

is not differentiable at

x=0

for any nonzero

k

because the left-hand derivative is

0

while the right-hand derivative is

k

. c)

h

is differentiable at

x=0

only when

k=1

. d)

h

is differentiable at

x=0

for every nonzero

k

because both pieces are polynomials and therefore smooth.