ASSIGNMENT 11
MAT 472 ·FALL 2005
Also look at: Problems 4–11 and 14 from Chapter V of Rosenlicht.
Problem 1 (Problem V.6).Suppose f:R→Ris differentiable on R. Prove that if there
exists M∈Rsuch that |f0(x)| ≤ Mfor all x∈R, then fis uniformly continuous on R.
Problem 2. Suppose that for each n∈N,fnis continuous on [0,1], differentiable on (0,1),
and satisfies fn(0) = 0. Also suppose that there exists M∈Rsuch that |f0
n(x)| ≤ Mfor all
n∈Nand all x∈(0,1). Prove that (fn) has a uniformly convergent subsequence.
Problem 3 (Problem V.9a).Use Cauchy’s Mean Value Theorem to prove the following
version of L’Hopital’s Rule: Let I= (a, b) be an open interval in R, and let fand gbe
differentiable on I, with gand g0nowhere zero on Iand limx→af(x) = limx→ag(x) = 0.
Then
lim
x→a
f(x)
g(x)= lim
x→a
f0(x)
g0(x),
if the second limit exists.
Problem 4 (See Problem V.11).Let fbe a real-valued function on an open set U⊆Rthat
is twice differentiable at x0∈U. Show that if f0(x0) = 0 and f00(x0)<0, then f(x0)> f(x)
for all x6=x0in some open ball centered at x0. (So x0is a local maximum of f.)
Date: October 31, 2005 / Due Date: Tuesday, November 8, 2005.
S. Kaliszewski, Department of Mathematics and Statistics, Arizona State University.
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