Two quadratic-type curves and are to be compared on the same set of axes.

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

Two quadratic-type curves

y=x^2

and

y=x^4

are to be compared on the same set of axes. (i) Show algebraically that for all real

x

with

|x|<1

x^4 < x^2

, and for all real

x

with

|x|>1

x^4 > x^2

. (ii) Using part (i), explain which curve lies closer to the

x

-axis near the origin and which one grows more rapidly for large

|x|

. (iii) Sketch both curves on the same set of axes, clearly indicating the regions identified in part (ii).