Consider the function

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

g(x)=\dfrac{x^{2}+1}{x-2}

, whose domain is all real numbers except

x=2

. (a) Compute

g'(x)

and determine every real number

c\ne2

for which either

g'(c)=0

g'(c)

does not exist while

g

is defined at

c

. List all such

c

–values. (b) Determine the sign of

g'(x)

on each interval defined by the critical numbers found in part (a) and by the vertical asymptote at

x=2

. Show clear analytical justification for the sign on each interval. (c) For every critical number listed in part (a), decide whether

g

has a relative maximum, a relative minimum, or neither at that number. Give the coordinate pair

(x,g(x))

for each relative extremum. (d) Explain how the presence of the vertical asymptote at

x=2

influences the existence or non-existence of relative extrema near that value.