Oxford University MAT 2016 Solved Papers

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Question : 8 of 10

Marks: +1, -0

Consider two functions

{c} f(x)=a-x^2

g(x)=x^4-a .

For precisely which values of

a>0

is the area of the region bounded by the

x

-axis and the curve

y=f(x)

bigger than the area of the region bounded by the

x

-axis and the curve

y=g(x)

?

all values of $a$ ,
$a>1$ ,
$a>\;{6}/{5}$ ,
$a>(\;{4}/{3})^{\;{3}/{2}}$ ,
$a>(\;{6}/{5})^4$ .

Solution:

The area bounded by the

x

-axis and the curve

y=f(x), A_1

is equal to

A_1=∫↙{-√{a}}^{√{a}} f(x) {d} x=\;{4}/{3} a^{\;{3}/{2}}

whilst the area bounded by the

x

-axis and the curve

y=g(x), A_2

is equal to

A_2=|∫↙{-\;^{4}√{a}}^{\;^{4}√{a}} g(x) {d} x|=\;{8}/{5} a^{\;{5}/{4}}

We require an

a

such that

A_1>A_2

, so

\;{4}/{3} a^{\;{3}/{2}} &>\;{8}/{5} a^{\;{5}/{4}}

20 a^{\;{6}/{4}} &>24 a^{\;{5}/{4}}

a^{\;{1}/{4}} &>\;{6}/{5}

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