8767 #63

Forum for the GRE subject test in mathematics.
MathCat
Posts: 171
Joined: Thu Oct 23, 2014 12:17 am

8767 #63

Can anyone explain the answer to #63 on 8767?

Let $f$ be a continuous, strictly decreasing, real-valued function such that $\int_0^\infty f(x)dx$ is finite and $f(0)=1$. In terms of $f^{-1}$ (the inverse function of $f$), $\int_0^\infty f(x)dx$ is...
(a) less than $\int_1^{\infty} f^{-1}(y)dy$
(b) greater than $\int_0^{1} f^{-1}(y)dy$
(c) equal to $\int_1^{\infty} f^{-1}(y)dy$
(d) equal to $\int_0^{1} f^{-1}(y)dy$
(e) equal to $\int_0^{\infty} f^{-1}(y)dy$

The answer is D.

antoniechan
Posts: 1
Joined: Thu Oct 23, 2014 4:12 am

Re: 8767 #63

just sketch the graph of the function

shade the region of the given integral

rotate your paper and you will see why the answer is D

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