Mathematics 242 – Principles of Analysis Problem Set 7 – due: Friday, April 4

‘A’ Section

1. Let f(x) = \frac{15x}{x^{4}+3x^{2}+1}. Use the Intermediate Value and/or Extreme Value Theorems to show the following: A) For all k ∈ [−3, 3], there exist c ∈ [−1, 1] such that f(c) = k. B) For all k with 0 < k < 3, there exist some c ∈ (1, ∞) such that f(c) = k. C) There is a c ∈ (0, 1) where f(c) = 3 D) There is a d ∈ (0, 1) where f′(d) = 0.

2. Show that there is a solution of the equation tan(x) = x in the interval (\frac{(2k−1)π}{2} , \frac{(2k+1)π}{2} ) for every k ∈ Z.

3. Using the definition of the derivative, find the value of f′(c), or say why f is not differentiable at x = c:

A) f(x) = x^{3} + 2x − 4 at c = 1. B) f(x) = sin(|x|) at c = 0. Hint: Look back at Problem Set 6, B 2. C) The function defined by

f(x) = \{ \begin{matrix}x^{2} & if x > 1 \\ 2x − 1 & if x ≤ 1 \end{matrix}

at c = 1. D) The function defined by

f(x) = \{ \begin{matrix}x^{2} & if x ∈ Q \\ 0 & if x ∈ Q^{c} \end{matrix}

at c = 0.

‘B’ Section

1. Let f be continuous on [0, 1] with f(0) < 0 and f(1) > 1. Suppose that g is another continuous function on [0, 1] such that g(0) ≥ 0 and g(1) ≤ 1. Show that there exists some c ∈ (0, 1) such that f(x) = g(x).

2. Let f be continuous on [a, b] with f(a) < k < f(b). Here is a variation on our proof of the Intermediate Value Theorem. A) Let

T = {x ∈ [a, b] | f(x) > k}.

Show that T has a greatest lower bound and that f(glb(T)) = k. B) Will this glb(T) always be the same as the c we found in our proof of the IVT with

f(c) = k? If so, prove they are the same; if not, give a counterexample.

3. This property deals with another property of real-valued functions of a real variable sometimes called Lipschitz continuity. A) Let f be a function on an interval I with the property that there exists a strictly

positive constant k such that |f(x) − f(x′)| ≤ k|x − x′| for all x, x′ ∈ I (this is the definition of Lipschitz continuity). Show that f is uniformly continuous on I.

B) The converse of the statement in part A is not true: Show that f(x) = x^{1/3} is uniformly continuous on [−1, 1], but there is no constant k such that |f(x) − f(x′)| ≤ k|x − x′| for all x, x′ ∈ [−1, 1]. Hint: Think slopes of secant lines to the graph y = x^{1/3}.