7. If f is C1 on [a, b] prove that there exists a cubic polynomial P such that f - P and its first derivative vanish at the endpoints of the interval.
8. If f ≥ 0, on [a, b] show that the polynomials approximating f may be all taken ≥ 0 on [a, b].
9. If f(c) = 0 for some point c in (a, b), prove that the polynomials approximating f on [a, b] may be taken to vanish at c.
10. Let f be an even function (f(x) = f(-x)) on [-1, 1]. Prove that if ∫_{-1}^1 f(x)x^{2k} dx = 0 for k = 0, 1, 2, ..., then f ≡ 0.
11. Prove that none of the power-series expansions of 1/(1 + x^2) converge to it on [-2, 2].
12. Let f be defined and C1 on (a, b), and suppose one-sided limits of f' exist at a and b. Prove that one-sided limits of f exist at a and b and f can be extended to a C1 function on [a, b].
13. Let Pn → f uniformly on [a, b] where {Pn} is a sequence of polynomials of degree ≤ N. Prove that f is a polynomial of degree ≤ N. (Hint: for each k ≤ N find a continuous function hk(x) such that ∫_a^b hk(x)x^j dx = 0 for all j ≤ N such that j ≠ k but ∫_a^b hk(x)x^k dx = 1, and consider lim_{n→∞} ∫_a^b hk(x)Pn(x) dx.)
14. a. For cm = ∫_{-1}^1 (1 - x^2)^m dx, obtain the identity cm = cm-1 - (1/2m)cm by integration by parts.
b. Show that
cm = 2 ∙ (2 ∙ 4 ∙ 6 ∙∙∙ (2m)) / (3 ∙ 5 ∙ 7 ∙∙∙ (2m + 1)) = (2(2^m m!)^2) / ((2m + 1)!)
15. Compute f * f for f equal to the characteristic function of [0, 1]
