Let 2= R, and suppose S₁ consists of intervals including Ø, the empty Ω set: S₁ = {(a,b]: -∞
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![(a) Let S = R, and suppose S₁ consists of intervals including Ø, the empty
set:
S₁ = {(a, b]: -∞0 ≤a ≤ b ≤∞0}.
If I₁, I2 € S₁, then I₁12 is an interval and in S₁ and if I € S₁, then Iº is a
union of disjoint intervals.
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- 4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard theorem for a suitable domain D. b. Then estimate the interval of existence of the solution.(a) Find a bijection f : Z → Z such that f^n is not equal to 1z for everyn ∈ N.(b) Find an injection f : R → R that is neither strictly increasing norstrictly decreasing.Find the open intervals on which r(t) = t3i +t4j is smooth.
- 4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard theorem for a suitable domain D. b. Then estimate the interval of existence of the solution. Use applied analysis and then I want the solution handwritten.A company has $300,000 to spend on two products. Suppose that theUtility derived by the company from x units of the first product and yunits of the second product is given byU(x,y) = 5000x0.75y0.25(i) Show that U is homogenous to some degree. (ii) Show that Euler’s Theorem holds for the function U(x,y)Let f: [a, b] -> R be continuous except on a finite number of points in (a, b]. Is f Lebesgue measurable? Answer it Yes or No. True or False: If f: E - › R is continuous and E has a Lebesgue measure 0, then its image f(E) has a Lebesgue measure 0.