MathDB

In the accompanying figure ,

y=f(x)

is the graph of a one-to-one continuous function

f

. At each point

P

on the graph of

y=2x^2

, assume that the areas

OAP

and

OBP

are equal . Here

PA,PB

are the horizontal and vertical segments . Determine the function

f

.
[asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("

y=x^2

",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("

y=2x^2

",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("

y=f(x)

",(24,576/10),W); label("

O(0,0)

",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("

P

",(20,400/25),E); label("

B

",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("

A

",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]

Problem Statement