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gbean
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Homework Statement
A function f is positive and increasing on [0, 1]. f(x) [tex]\leq[/tex] 1 [tex]\forall[/tex] x in [0, 1]. Show that f^2 is integrable, and that [tex]\int[/tex] f^2(x) dx [tex]\leq[/tex] [tex]\int[/tex] f(x) dx.
Homework Equations
The Attempt at a Solution
Since f is increasing and positive, it is also monotone. If f(x) is monotone on [a, b], then f(x) is integrable on [a, b]. Also, when x<y, then f(x) < f(y).
f^2(x) - f^2(y) = (f(x) - f(y))(f(x) + (f(y))
Since f is increasing, (f(x) - f(y)) < 0 and (f(x) + f(y)) > 0.
Apparently, the last 2 lines are supposed to show that f^2(x) is increasing, but I don't understand the reasoning.
And if I show that f^2(x) is increasing, then it is also integrable, which solves one part of the question.
Then since f^2(x) [tex]\leq[/tex] f(x) for x>0, and f(x)[tex]\leq[/tex] 1 by the given => [tex]\int[/tex] f^2(x) dx [tex]\leq[/tex] [tex]\int[/tex] f(x) dx, by inequality of functions indicating also the inequality of the integrals. This solves the second part of the question.