Question

Show E[f(X)g(X)]≥E[f(X)]E[g(X)] for f,g bounded, nondecreasing.

Answer #1

Show that if f is a bounded function on E with[ f]∈ Lp(E), then
[f]∈Lq(E) for all q > p.

Let f be a bounded measurable function on E. Show that there are
sequences of simple
functions on E, {(pn) and {cn}, such that {(pn} is increasing and
{cn} is decreasing and each of
these sequences converges to f uniformly on E.

1. Show that U=f(x) + e^(-3x) g(2x+y), where f and g are
arbitrary smooth functions, is a general solution of Uxy- 2Uyy
+3Uy=0. Do not solve!

(a) Show that the function f(x)=x^x is increasing on (e^(-1),
infinity)
(b) Let f(x) be as in part (a). If g is the inverse function to
f, i.e. f and g satisfy the relation x=g(y) if y=f(x). Find the
limit lim(y-->infinity) {g(y)ln(ln(y))} / ln(y). (Hint :
L'Hopital's rule)

Compute the area of the region bounded by the curves
f(x)=1/x+2,g(x)=(x+2)^2 and the lines x=-3/2 and x=1

Let f(x) g(x) and h(x) be polynomials in R[x].
Show if gcd(f(x), g(x)) = 1 and gcd(f(x), h(x)) = 1, then
gcd(f(x), g(x)h(x)) = 1.

Find the area of the region bounded by the two functions f(x) =
x^2 − 4x and g(x) = 4x.

Find the area bounded between the graphs of f(x) = x3 + 11x2 −
11 and g(x) = 4x2 − 8x + 5. (A) 625/12 (B) 104/5 (C) 233/4 (D)
113/6 (E) 82/3

Let f: R --> R be a differentiable function such that f' is
bounded. Show that f is uniformly continuous.

Assume f : [a, b] → R is integrable. (a) Show that if g
satisfies g(x) = f(x) for all but a finite number of points in [a,
b], then g is integrable as well. (b) Find an example to show that
g may fail to be integrable if it differs from f at a countable
number of points.

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