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- Expand x^2y+3y-4 about the point (-1,2) by Taylor's theoremA company has $300,000 to spend on two products. Suppose that the Utility derived by the company from x units of the first product and y units of the second product is given by U(x,y) = 5000x^0.75y^0.25 (i) Show that U is homogenous to some degree. (ii) Show that Euler’s Theorem holds for the function U(x,y)The functions f(x, y) = x 2 + y 2 and g(x, y) = x 2 - y 2 both have acritical point at (0, 0). How is the behavior of the two functions at the critical point different?
- . Show that it is possible to solve u and v from:xyu + xy2uv2 = 2 and x2yvu4 + yu3v2 = 2,in terms of x and y uniquely near the point (x, y, u, v) = (1, 1, 1, 1); also find the first four partial derivatives at the point (1, 1).2. Give an example showing that, even if f0(c) = 0, it could happen that f(x) does not have a localmaximum or minimum at x = c. How is this consistent with Fermat’s Theorem in the book?a) f(x) = x2 sin x is restricted to the domain 10 ≥ x ≥ 0 and it has four critical points. They are at x = 0 and x = 2.89 and x= 5.087 and x = 8.096 At what value x in this domain does f(x) attain its maximum? b) What is the minimum value of f(x) = x^2 - 8 ln x on the interval [1,5]
- From the partial differential eqiuation by eliminating the arbitrary function z=(x2+y2+z2)Suppose U solves the heat equation on the real lineUt = 4Uxx, x ∈ Rwith initial valueU(x, 0) = (4, x ≤ 02, x > 0.(i) Use the Fourier-Poisson formula to give an explicit expression for the solutionU.(ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot outthe solution at several instants of time to explain your answer. What is the limitof U as t → ∞?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)