Consider the given LP: Maximize z = 4x1 + 6x2 + 8x3 Subject to        3x1 + 2x2 + 5x3<= 30        9x1 + 2x2 + 7x3 <= 126        2x1 + 3x2 + x3<= 60        xi>= 0 1-Find the optimal values of Z, x1, x2, x3 by using the simplex method. 3-If the RHS of the constraint 1 is 34 instead of 30, what is the new value of Z?

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Show that Jσ_(ij)D_(ij)=T_(iL)F_(iL˙)=S(AB)E_(AB˙) [Hints: (a) to prove the first equation, use the relationship between F_(iL)˙ and L_(ij), and expand L_(ij) into symmetric and antisymmetric parts. Note the symmetry of σ_(ij).  (b) To prove the second equation, first express E_(AB)˙ using F_(iL), then take the time derivative. Substitute into the definition of S_(AB )in index form and simplify.]

Calculate the Fourier transform of f(t)=3sin2t-2sint-cost and plot the spectral coefficients.

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Use the following theorem. = If F(s) L{f(t)} and n = 1,2,3, dots, n n then L{i" f(1)} = (- 1)" d² _F(s). ds n Evaluate the given Laplace transform. (Write your answer as a function of s .) L{4t cos(t)}

2 Vidhya Balan is planning to liquidate her investments inmutual funds and invest in real estate. Before makingthe change to her investment strategy, Vidhya wants toforecast the price of mutual funds for the next 2 months.She has collected the following data on the average fundprices for the past 10 months.Month                 Average                         Fund Price1                           55.12                           53.83                           53.44                           52.955                           52.156                           52.757                           52.658                           51.59                            52.2510                           51.7a Using a five-period moving average, forecast theaverage fund price for Period 11.b. Using exponential smoothing with α = 0.3 forecastthe average fund price for Period 11. Assume an initialforecast for Month 2 (F2) as 55.10

2. Suppose f(x)=7x-2 is a function. Find the domain and range of f(x).

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