1. [3 points] Outline a proof, beginning with basic properties of the real numbers, of the following theorem - if f : [a, b] → (−∞, ∞) is a continuous function such that f J(x) = 0 for all x ∈ (a, b), then f (a) = f (b).
2. [3 points] Suppose that f is a positive differentiable function on (0, ∞). Prove that: (i) if f J is finitely and is nonzero for each x ∈ (0, ∞).
3. [3 points] Let f : [0, 1] → (−∞, ∞) be continuously differentiable and obey the ordinary differential equation f (t) = e f (t). Show that: (i) if 0 < t0 < 1 then f cannot have a positive local maximum at t0; (ii) if 0 < t0 < 1 then f cannot have a negative local minimum at t0; (iii) if f (0) = f (1) = 0 then f (t) = 0 for all t ∈ [0, 1].
4. [3 points] Give an example of a function f : (−∞, ∞) → (−∞, ∞) having all three of the following properties: (i) f (x) = 0 for x < 0 and x > 2; (ii) f (1) = 1; (iii) f has derivatives of all orders.
5. [3 points] Show that if n is a natural number and α, β are real numbers with β > 0 then there exists a real function f with derivatives of all orders such that: (i) |f (k)(x)| ≤ β for k ∈ {0, 1, ..., n − 1} and x ∈ (−∞, ∞); (ii) f (k)(0) = 0 for k ∈ {0, 1, ..., n − 1}; (iii) f (n)(0) = α.
6. [3 points] Suppose that y = f (x) : (−∞, ∞) → (−∞, ∞) is infinitely differentiable and has a local minimum at 0. Prove that there exists a disc centered on the y axis which lies above the graph of f and touches the graph at the point (0, f (0)).
7. [3 points] Suppose that f is continuous function on (−∞, ∞) which is periodic with period 1, i.e., f (t + 1) = f (t) for all t ∈ (−∞, ∞). Prove that: (i) f is bounded and achieves its maximum and minimum; (ii) f is uniformly continuous on (−∞, ∞); (iii) there is a t0 ∈ (−∞, ∞) such that f (t0 + π) = f (t0).
8. [3 points] A function f : [0, 1] → (−∞, ∞) is said to be upper semi-continuous if given x0 ∈ [0, 1] and ϵ > 0, there is a δ > 0 such that if |x − x0| < δ, then f (x) < f (x0) + ϵ. Show that an upper semi-continuous function f on [0, 1] is bounded above and attains its maximum value at some point x∗ ∈ [0, 1], and answer the question of whether or not there is a corresponding result on the so-called ‘lower semi-continuous function’.
9. [3 points] Suppose that (a ) is a sequence of positive numbers. Show that: (i) ∞ a <
∞ √a a < ; (ii) ∞ √a a < ∞ a < ; (iii) if ∞ a < then there is a sequence of positive numbers (cj) such that limj→∞ cj = 0 and Σ∞j=1 cjaj < ∞.
10. [3 points] Evaluate two limits: (i) lim cos π cos π • • • cos π ; (ii) lim ( 1 + 1 +• • •+ 1 ).
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