Mathematics

Problem

Let

f(x)=\begin{cases} x^{2}, & x\le 1,\\ 3x-2, & x>1. \end{cases}

Determine whether

f'

exists at

x=1

. Which statement is correct? a)

f'

exists at

x=1

because the function is continuous there. b)

f'

does not exist at

x=1

because the left- and right-hand slopes are different. c)

f'

exists at

x=1

because both one-sided limits of the difference quotient equal

2

. d)

f'

does not exist at

x=1

because

f

has a vertical tangent there.