Let f(x)=\int_0^1(x t)^2 {d} t, \text{ and } g(x)=\int_0^x t^2 {d} t . Let A>0. Which of the following statements is true?

Correct Answer is : f(g(A)) is always bigger than g(f(A)).

g(f(A)) is always bigger than f(g(A)).

f(g(A)) is always bigger than g(f(A)).

Oxford University MAT 2015 Solved Papers

Question : 4 of 10

Marks: +1, -0

Let

f(x)=∫↙0^1(x t)^2 {d} t, \;\; \;\text " and "\; \;\; g(x)=∫↙0^x t^2 {d} t .

Let

A>0

. Which of the following statements is true?

Solution:

Evaluating the integrals

f(x)=\;{x^2}/{3}

and

g(x)=\;{x^3}/{3}

. Hence

g(f(A))=\;{A^6}/{3^4}

and

f(g(A))=\;{A^6}/{3^3}

. Hence the answer is (b).

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