Question

You are having a meeting with the CEO of a soda company. You have interpreted the number of cans of soda produced versus profit as the function P(x) = x4 − x3 − 11x2 + 9x + 18. Describe to the CEO what the graph looks like and, in general, how to sketch the graph without using technology. Use complete sentences, and focus on the end behaviors of the graph and where the company will break even (where P(x) = 0)

Answer 1

The shape of the graph is similar to the letter W.

$P(x)=0$ at points $x=-3;-1;2;3.$

At intervals:

$x=-\infin...-2.75$ the function graph descreasing from $\infin\space to -21$ ;

$x=-2.75...0.5$ the function graph increasing from $-21\space to\space 20$ ;

$x=0.5...2.5$ the function graph descreasing from $20\space to\space -5$ ;

$x=2.5...\infin$ he function graph increasing from $-5\space to\space \infin$ ;

End behavior: $x\rarr\infin,P(x)\rarr\infin$ .

The region of interest will lie on the right of the origin, which means only

the right half of the W(a negative number of cans cannot be produced).

 The company will be at a loss if the number of cans produced is near $2.5$.

The break even situations are present if the company

produces 2 or 3 cans $(P(x)=0)$. The profit produced will be maximized

if the number of cans produced is

greater than 3.