$()$ Dr$.$ Bob sets up a new company, Alpha Electronics, producing specialized but low tech computer chips. The demand $q$ in thousands of chips per month is related to the price $p$ in dollars by

$q(p)=20e^{-p}$

Note that there is still demand $q>0$ even for very large prices $p($some consumers need these chips no matter the cost$).$ The chip factory has fixed costs of $$1,000$ per month and each chip costs $$1$ to manufacture.

$($a$)(1$ mark$)$ Find the inverse of the demand curve, the function $p(q).$

$($b$)(2$ marks$)$ Find the profit function $P(x)$ in thousands of dollars per month where $x$ is the production of the factory in thousands per month matching demand $)=(x.$

$($c$)(2$ marks$)$ Find the maximum profit per month and the corresponding production $x.$ Give a short justification of how you know this is the maximum. producing $y$ thousand chips per month.

$($a$)(1$ mark$)$ Find $P(x,y)$ and $Q(x,y),$ the profit functions for the Alpha and Beta factories, respectively. Note that the price the chips can be sold at corresponds to the demand of the combined production $q=x+y.$

$($b$)(2$ marks$)$ If the two companies agree to produce the same amount $x=y$ what is the maximum profit they can each achieve and the corresponding production $x?$

$($c$)(1$ mark$)$ Explain in words $($one sentence$)$ why the combined profit of the two companies is lower than the maximum profit of one producer from question #$2.$ Electronics realizes that they can increase profits by changing their production $y.$ Note that this question will require a calculator or other computational tools for numerical answers.

$($a$)(2$ marks$)$ Write $Q(y)$ in this situation and calculate $Q^{\text{'}}(y).$

$($b$)(2$ marks$)$ Find the appropriate critical point for $Q$ using Newton's method. Show the iteration formula, your initial estimate of the root of $Q^{\text{'}},$ and the next three iterates. Also show the critical point value that you believe is accurate to four decimal places.

$($c$)(2$ marks$)$ Compute the optimal profit for Beta Electronics in this scenario. What is the profit for Alpha Electronics now $($still producing at $x$ from the previous question$)?$