Mathematics

Problem

Consider the piecewise–defined function

f(x)= \begin{cases} x^{2}-1, & x<3,\\ mx+b, & x\ge 3, \end{cases}

where

m

and

b

are real constants. a. Part 1: (a) Compute

\displaystyle \lim_{x\to 3^-} f(x)

and

\displaystyle \lim_{x\to 3^+} f(x)

in terms of

m

and

b

. Show all work. b. Part 2: (b) For which values of

m

and

b

f

continuous at

x=3

? Justify your answer. c. Part 3: (c) Compute

f'(x)

for

x<3

and for

x>3

. d. Part 4: (d) Determine all values of

m

and

b

for which

f

is differentiable at

x=3

. Provide a complete justification that addresses continuity and the equality of the one–sided derivatives.