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Homework 10 Calculus 1
1. Recall that a function is convex if for all a < b, we have
f (λa + (1 − λ)b) ≤ λf (a) + (1 − λ)f (b) ∀λ ∈ [0, 1].
(The inequality says that over any interval [a, b], the graph of f (x) is below the line connecting (a, f (a) and (b, f (b)). See the hint of Homework 8 for an illustration.) Prove that if a function f : R → R is convex, then the graph y = f (s) is always above any of its tangent line. That is, if P (x) is the equation of tangent line to y = f (x) at x = a, then f (x) ≥ P (x) for all x.
2. Rudin Chapter 5, Problem 14. (You might find Problem 1 useful.)
3. Rudin Chapter 5, Problem 15. (Don’t worry about the ”To show...” at the end of the problem.)
4. Rudin Chapter 5, Problem 17. (Hint: first show that P0 = P1 = 0, and therefore P2 = ax2.)
5. Rudin Chapter 5, Problem 25 part (a) - part (c). (For part (b), the result follows easily from (a) if we know that xn is convergent, why? For part (c), apply Taylor’s Theorem at xn. You might find Problem 2 might be useful.)
6. Rudin Chapter 5, Problem 26. (Start with |f (x)| = |f (x) − f (a)| = · · ·.) 7. Salas 4.11: 17, 20.
8. Salas 11:5: 10, 28, 48.
9. Salas 11.6: 15, 28, 42.
10. Salas 12.6: 12, 16, 18, 22, 32.
11. Salas 12.7: 10, 22, 29.
