Define the piecewise function p(x)= cases x^ 2 -4 x-1 , & x 1, 0, & x=1

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

Problem

Define the piecewise function

p(x)=\begin{cases}\dfrac{x^{2}-4}{x-1}, & x\ne1,\\0, & x=1.\end{cases}

Consider the closed interval

[-2,4]

. (1) Part 1: (a) Show that

p

is continuous on

[-2,4]

. (2) Part 2: (b) Compute

p''(x)

for

x\ne1

and simplify your result. (3) Part 3: (c) Determine all

x

[-2,4]

at which

p''(x)=0

p''(x)

is undefined. (4) Part 4: (d) Create a complete sign chart for

p''

[-2,4]

and hence determine the intervals on which the graph of

p

is concave up and concave down. (5) Part 5: (e) State the

x

-coordinates of any points of inflection of

p

[-2,4]

and justify each using your work above.