MATH 295 PROBLEMS 14

IGOR KRIZ

1. Let a > 0. Prove that the maximum value of the function f : ℝ → ℝ given by

f(x) = \frac{1}{1 + |x|} + \frac{1}{1 + |x − a|}

is (2 + a)/(1 + a). [Hint: Derivative can be found on each of the intervals (−∞, 0), (0, a) and (a, +∞).]

2. Sketch a graph of the function f : ℝ → ℝ given by

f(x) = \frac{1}{12}x^{4} − \frac{1}{2}x^{3} + x^{2} + x + 1,

showing regions of convexity, concavity and inflection points. [Do not use a calculator.]

3. Prove that if f is a funcion defined on an interval I, then f is convex if and only if for all x, y ∈ I and for every t ∈ [0, 1] we have

f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y).

[Hint: it is just a restatement of the definition.]

4. Prove that the Newton method always converges for an increasing convex function f : ℝ → ℝ for which there is an x ∈ ℝ with f(x) = 0. In more detail, show that if we select a_{1} ∈ ℝ arbitrarily, and define a_{n}, n = 2, 3, 4, ... inductively by

a_{n+1} = a_{n} − \frac{f(a_{n})}{f′(a_{n})},

then

lim_{n→∞} a_{n} = x.

[Hint: Look for discussion of Newton method in class or in the book.]

5. Suppose that f is a real function defined on an interval J and suppose that there exists an n > 1 such that |f(x) − f(y)| ≤ |x − y|^{n} for all x, y ∈ J. Prove that then f is constant. [Hint: consider its derivative.]