Mathematics

Problem

Consider the function

p(x)=\begin{cases} mx+1, & x\le 0,\\ x^2+n, & 0<x<4,\\ \sqrt{x+12}, & x\ge 4. \end{cases}

The constants

m

and

n

are real numbers. a. Part 1: (a) Write an equation in

m

that results from imposing continuity of

p

x=0

. Solve for

m

. b. Part 2: (b) Write an equation in

n

that results from imposing continuity of

p

x=4

. Solve for

n

. c. Part 3: (c) Verify explicitly that the values of

m

and

n

found in parts (a) and (b) make

p

continuous on the entire real line. Your verification should reference each relevant one–sided limit and function value.