PROBLEM SET 6 (DUE IN LECTURE ON OCT 23 (FRIDAY))
(All Theorem and Exercise numbers are references to the textbook by Apostol; for instance
“Exercise 1.15-3” means Exercise 3 in section 1.15.)
Problem 1. The following functions are defined on R. For each function, determine at
which points in R the function is differentiable and compute the derivative at
those points.√
(a) f (x) = 1 + x2
(b) f (x) = x · |x|
(c) f (x) = xm (x√
+ 1)n , where m, n ∈ N
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(d) f (x) = (1 + x)3
Problem 2. Let n be a positive integer and let f1 , . . . , fn : S → R be differentiable functions with the same domain. Let g = f1 · · · fn be the product of these functions. Prove that for any x such that g(x) 6= 0, we have
g 0 (x)
f10 (x)
fn0 (x)
=
+ ··· +
.
g(x)
f1 (x)
fn (x)
Problem 3. Let f : R → Z be an integer-valued function. Show that if f is differentiable
at x ∈ R, then f 0 (x) = 0.
Problem 4. Suppose f : R → R is an even function (that is, f (−x) = f (x) for all x ∈ R)
and is differentiable at 0. Prove that f 0 (0) = 0.
Problem 5. (a) Construct a differentiable function f : R → R such that f (x) = 0 for
x ≤ 0, f (x) = 1 for x ≥ 1, and 0 < f (x) < 1 for 0 < x < 1. (Hint: Define
f on the interval (0, 1) as an appropriately chosen cubic polynomial.)
(b) Use the function f constructed in the previous part to define a new differentiable function g : R → R such that g(x) = 0 for |x| ≥ 1, g(0) = 1,
and 0 < g(x) < 1 for 0 < |x| < 1. (Hint: multiply together two functions
of the form f (ax + b).)
(c) Let h : R → R be a continuous function that is differentiable at all points
except 0. Prove that there exists a function j : R → R, differentiable on
all of R, such that j(x) and h(x) have the same sign (both positive, both
negative, or both zero) for all nonzero x and j(x) = h(x) for |x| ≥ 1.
(Hint: multiply h by an appropriate modification of g.)
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