Use the method of lagrange multipliers to find the maximum value of f(z, y) ay for positive values of x and y, subject to the constraint x + 4y = 12. 4. suppose that p(x, y) = 2x1/3y2/3 units of a product can be made using r units of labor and y units of capital. suppose that the cost per unit of labor is $16 and the cost per unit of capital is $4. use the method of lagrange multipliers to find the amounts z and y of labor and capital, respectively, that minimize the cost of producing 32 units of product. verify that for the values of r and y that minimize cost, 16

Have you ever been in the middle of telling a story and had to use hand gestures or even a drawing to make your point? in algebra, if we want to illustrate or visualize a key feature of a function, sometimes we have to write the function in a different form—such as a table, graph, or equation—to determine the specific key feature we’re looking for. think about interactions with others and situations you’ve come across in your own life. how have you had to adjust or change the way you represented something based on either the concept you were trying to describe or the person you were interacting with? explain how the complexity or layout of the concept required you to change your style or presentation, or how the person’s unique learning style affected the way you presented the information.