1. A company is analyzing the effectiveness of its marketing strategy for launching a new product. The company finds that the number of new customers N(t) projected to buy its new product can be represented by the function: N(t) = 15 + 9t² - 1.5t³ where "t" is number of years after launching the new product, and N(t) is the number of new customers (measured in 1000's) at time "t"

a) How many years "t" after launching new product will number of new customers N(t) reach maximum?

Solve using "Optimization Procedure based on Second Derivative Test" Show All Work! Hint: Find First and Second derivatives; then find and classify critical point(s) of N(t) function.

b) What is the maximum number of new customers N(t) projected to buy the new product after the launch of new product? (Hint: Evaluate N(t) at the optimal "t" found above.)

2) An online company sells a cordless hedge-trimmer with the following price-demand relation: p = 540 x^-1/2 where "p" is price in dollars and "x" is weekly demand. The company's total weekly Cost associated with the cordless hedge-trimmers is given by: C(x) = 45x + 900

a) Find the Revenue function for the cordless hedge trimmer, and simplify the function.

b) Find the Profit function for the cordless hedge trimmer, and simplify the function.

c) How many cordless hedge-trimmers must the company sell per week to Maximize (optimize) weekly Profit? Solve using the "Optimization Procedure based on Second Derivative Test" Show All Work! Hint: Find First and Second Derivative, then find and classify the critical point(s) of Profit function.

d) Find the selling price "p" that corresponds to the Maximum Profit.