Note: Yes, this is homework. But no, I am not intending to copy straight the answers from here, however if you can work me through most of it (as opposed to just hints) that would be helpful. This way, I can take your lead, work my way though it, and have help should I get stuck again. Thank you!
8. Suppose that f is differentiable with derivative f '(x) = (1 + x^3)^(-1/2). Show that g = f ^ (-1) satisfies g''(x) = (3/2)*g(x)^2.
22. a. Prove that an incresing and decreasing function intersect at most once.
b. Find two continuous increasing functions f and g such that f(x) = g(x) precisely when x is an integer.
25.b. Prove that if a function is differentiable and nondecreasing, then f'(x) >= 0 for all x.
c. Prove that if f'(x) >= 0 for all x, then f is nondecreasing.
13. Supopse that f is a continuous increasing function with f(0) = 0. Prove that for a,b > 0 we have:
a*b =< integral (0 to a) f(x)dx + integral (0 to b) f ^ (-1) (x)dx
and that equality holds if and only if b = f(a).
2006-11-11
11:39:00
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3 answers
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asked by
Andrew H
1
in
Mathematics