1.1 1.2 Show that if w is continuous on [0, 1] and jwvdx=0 VveV, ic. p / then w(x)=0 for xe[0, 1]. Show that under suitable assumptions the problem B above can be given the formulation (1.1).
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- Any function of the form f(x, y) = ax*y" where a is a real number and k, n are positive integers satisfies the equation fay – fyæ = 0 - True False□ 5. Prove that 2x² – 8x = = IVT. 22 has at least one soution on the interval [5,10] by usingConsider a dataset of numbers y1, y2, ..., yn and the linear function f(x) = 9x − 4. Prove or find a counterexample to disprove. If a ≤ b, show that f(a) ≤ f(b). since it is a monotonic function that is increasing i was thinking of doing the derivative to prove it but i dont what else to do more to prove it after doing it.
- 18. Let f(x) be continuous on [1, 3] and f (1) = 4, f(3) = 20. Show using the intermediate value theorem that the equation f(x) = 5x has at least one solution in the interval (1, 3). (Hint: Consider the function g(x) = f(x)-5x.) %3D %3D %3D31,375. (S.10). By using the second divided difference find f [1, 1.2, 1.4] for f (x) = e=*² 6. (S.15). Let f (x) using the Three-Point formulas to approximate f(1) (1); Vx3 + x² + 2 , xo = 1, h = 0.5, x1 = 1.5, x2 = 2. By
- 10.2.C. (3rd bullit point). Thanks.3.4.1. Using Definition 3.35, prove that each of the following functions is uni- formly continuous on (0,1). a) f(x) = x² + x b) f(x) = x³ = x + 2 c) f(x) = x sin 2x2.6. Let f be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let %3D 1 fa+1(x) п! n = 0, 1, 2, .. (x - a) Show that the nth derivative of f, exists and equals f. b) Prove the following theorem of M. Fekete: The number of changes in sign of f in [0, a] is not less than the number of changes in sign in the ordered set of numbers f(a), f1(a), . .., f,(a). Hint. Use mathematical induction.
- Ex 9.5: Prove: if f(z) is analytic and real on a domain R then I is a constant function. Pf: I is analytic and real on √2 => f(x, y) = u(x, y) + i.0₂ namely √(x, y) =0; CR equations hold for every ZER I from analyticity on √}: OV 04 = 0 and -0 =0 OX by ду ди Du DX = By = 0 => M(x,y) = CEⓇR =) f(x,y) = c, namely & is a constant function. D -Q.2 A) Find a, b if f(x) is continuous at x=3, f (-1) =6 ? Let f(x) = + a, x # 3 x-3 ax +3b, x = 3 B) Is the f(x) is continuous at x=2 ? (x³-8 x >0 Let f(x) = 8. x-25. Let f be defined for all real numbers and suppose if(x)-f(y)| ≤ (x - y)² for all real numbers, x and y. Prove f'(y) = 0, vy E R.