Mathematics 242 – Principles of Analysis Problem Set 7 – due: Friday, April 12

’A’ Section

1. Let f(x) = \frac{15x}{x^{4}+x^{2}+1}. Using the Intermediate Value Theorem, A) Show: For all k ∈ [−5, 5], there exist c ∈ [−1, 1] such that f(c) = k. B) Show: For all k with 0 < k < 5, there exist some c ∈ (1, ∞) such that f(c) = k. C) Show that if m satisfies 0 < m < 15, then f(x) = mx has two real solutions other

than x = 0.

2. Show that there are at least three real solutions of the equation sin(x) + 2 cos(x) = x/2. Hint: Look at the values of g(x) = sin(x) + 2 cos(x) − x/2 at “nice” multiples of \frac{π}{2} .

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 + 1 at c = 2. 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.

4. Suppose f, g are differentiable functions with f(g(x)) = \frac{x}{x^{2}+1} and that

\begin{matrix}g(1) = 1 & g(2) = 4 \\ g′(1) = 2 & g′(2) = −1 \end{matrix}

Determine the equation of the tangent line to the given graph at the given point. A) y = f(x) at (1, f(1)). B) y = (f ∘ g)(x) at (2, (f ∘ g)(2)).

’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}.

4. Let f and g be differentiable on (a, c) and let b ∈ (a, c). Assume f(b) = g(b). Define a new function by

p(x) = \{ \begin{matrix}f(x) & if x ∈ (a, b) \\ g(x) & if x ∈ [b, c) \end{matrix}

Show that p is differentiable on (a, c) if and only if f′(b) = g′(b).