Use the method of Lagrange multipliers to optimize the function subject to the given constraint. Find the maximum value of the function f(x,y,z) = x+3y- 5z subject to the constraint z = 2x^2 + y^2.

Fit a linear function of the form f (t) = c0 + c1t to the data points (0, 0), (0, 1), (1, 1), using least squares. Use only paper and pencil. Sketch your solution, and explain why it makes sense.

A collector of antique clocks sold at auction believes that the price received for the clocks may be modeled as a linear function of the age of the clocks and the number of bidders at the auction. After a preliminary study and collection of 32 observations, the collector came up with the following second-order model: y = BO+B₁x₁+B₂x₂ + ß3 x1x2 +B4x² +ɛ, where y: price of a clock; x₁: age of a clock; x2: the number of bidders. Considering the output below and by setting a = 0.05, answer the following questions. The regression equation is Price = 262 +2.26 Age + 14.2 Bidder + 1.13 AgeBid 4.20 Bid^2 Predictor Constant -261.7 2.260 14.21 Age Bidder AgeBid Bid^2 S = 84.512 Coef SE Coef 1.1301 -4.196 Analysis of Variance Source 404.4 2.052 R-Sq= 96.0% Variable Age Bidder AgeBid Bid^2 60.83 0.23 5.17 0.000 0.219 1.344 -3.12 0.004 R-Sq (adj) DF Regression 4 4606950 Residual Error 27 192840 Total 31 4799790 SS T P -0.65 0.523 1.10 0.280 0.817 a. Is the model useful as a whole? Apply an…

Solve Prob. 7 if the constraint is changed to y ≥ −2. Prob. 7 Find the minimum of the function in y ≥ 0. Verify the result analytically.