1. Suppose a consumer seeks to maximize the utility function U (1,₂)= subject to the budget constraint - x1 X2 P11+ P₂x₂ = Y, where ₁ and ₂ represent the quantities of goods consumed, p₁ and p2 are the prices of the two goods and Y represents the consumer's income. (a) (10) What is the Lagrangian function for this problem?

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(a) Suppose we create a model of the height of a shrub (Y) based on the amount of bacteria in the soil (X1) and whether the plant is located in partial or full sun (X2). Height is measured in cm, bacteria is measured in thousand per ml of soil (so 1000 is 1 unit), and type of sun = sun and type of sun = between shrubs with a 5500 bacteria and those with a 4000 bacteria count. Assuming the sun is the 0 if the plant is in partial 1 if the plant is in full sun. What will be the height difference on average same for both. (b) Let's say after a few training steps your model is in this state: Y = 33 + 4.6 * X1+8* X2. Explain what the effect of the plant being in sun vs. partial shade will have on the average height of the shrub if there are no bacteria in the soil, based on the model.

1. We found that the marketing research department for the company that manufactures and sells memory chips for microcomputers established the following​ price-demand and revenue​ functions:   p(x)=90−3x ​Price-demand function R(x)=xp(x)=x(90−3x) Revenue function   where​ p(x) is the wholesale price in dollars at which x million chips can be​sold, and​ R(x) is in millions of dollars. Both functions have domain 1≤x≤18.   A) Which of the following is the graph of the revenue​ function? (B) Find the output that will produce the maximum revenue. ​(C) What is the wholesale price per chip that produces the maximum​ revenue?   I stuck at the part where = −3​(2x^2+90​x)     I am supposted to add number in to complete the square inside the parentheses and add outside the parenthes? what I don't what the number is and how to get it? can explain that part to me. Thank You