Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L? Let f be defined on an open interval I containing a point a...
9 3. Given the sample (11, 12). Prove that H = A where H, G and A are harmonic mean, geometric mean and arithmetic mean respectively [5 marks]
[3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that \f'(x) < 1/(1 - 1z| for all z e D[0, 1]. [3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that f'(x) < 1/(1-1-12 for all z e D[0, 1]
Let f(3) = 1 (a) Prove {f} 1 + nx converges to 0 pointwise on (-0,00). (b) Prove or disprove {n} , converges to 0 uniformly on (-0, 0);
31 (a) If fis integrable, prove that fa is integrable. Hint: Given e>0, let h and k be step functions such that h f k and j (k-h) < ε/M, where M is the maximum value of Ik(x) +h(x)]. Then prove that h and k2 are step functions with h' srsk (we may assume that OShSSk since f is integrable if and only if I is-why?), and that I (k2 - h2) <e. Then apply Theorem 3.3. (b) If fand...
(3) 5. Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that f'(2) < 1/(1 - 121) for all z e D[0,1].
Prove that (f = O(g)] ^ (g = O(h)] = f = O(h).
Real analysis 7. Assume that f and g are differentiable functions such that f(0) 9(0) and that for all & ER, S' () > '(x). Prove that f(c) > 9() for all > 0.
9. Given any nonconstant polynomial f(x) with integral coefficients, prove that there are infinitely many primes p such that f(x) = 0 (mod p) is solvable. (H)
Find f(a), f(a+h), and the difference quotient f(a+h)-f(a)/h, where h ≠ 0f(x)=9-2x+8x²f(a)= f(a+h)= f(a+h)-f(a)/h=