MathDB

(1) Let

x>0,\ y

be real numbers. For variable

t

, find the difference of Maximum and minimum value of the quadratic function

f(t)=xt^2+yt

0\leq t\leq 1

.
(2) Let

S

be the domain of the points

(x,\ y)

in the coordinate plane forming the following condition:
For

x>0

and all real numbers

t

with

0\leq t\leq 1

, there exists real number

z

for which

0\leq xt^2+yt+z\leq 1

.
Sketch the outline of

S

.
(3) Let

V

be the domain of the points

(x,\ y,\ z)

in the coordinate space forming the following condition:
For

0\leq x\leq 1

and for all real numbers

t

with

0\leq t\leq 1

0\leq xt^2+yt+z\leq 1

holds.
Find the volume of

V

.
2011 Tokyo University entrance exam/Science, Problem 6

Problem Statement