Question

You are having a conference call with the CEO of a paper company. You have interpreted the number of trees cut down versus profit as the function P(x) = −x4 + x3 + 7x2 − x − 6. 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). (10 points)

Answer 1

You can only cut down a integer number of trees. So you might look at a few integer values for x. As x get large the –x4 term dominates the expression for big losses. x = 0 is easy P(x) = -6. Without cutting any trees you have lost money Put x = 1 and you get for the terms in order  -1 + 1 + 7 -1 -6 = 0. So P(x) crosses zero just before you cut the first tree. So you make a profit on only 1 tree. However when x=10 you are back into no profit. So compute a few values for x = 1,2,3,4,5,6,7,8,9.

Answer 2

Answer:  

Given Function is,

$P(x) = -x^4 + x^3 + 7x^2 - x - 6$

Which is the bi quadratic function.

For x-intercept, P(x)=0

Thus, $-x^4 + x^3 + 7x^2 - x - 6=0$

⇒ $(x-1) ( -x^3+7x+6)=0$

⇒ $(x-1) (x+1) (x+2) (x-3) = 0$

⇒ The x-intercept of the functions are, (-2,0),  (-1,0), (1,0) and ( 3,0)

Similarly, for y-intercept,

x = 0 , y = -6

Thus, y-intercept of the given function is,  (0,-6)

The critical points of the function f(x) are ( where f'(x) = 0), (2.254, 12.949)

and (-1.574, 2.879)

End Behavior:

Since, The leading coefficient is negative and has even degree,

Therefore, end behavior of the given function is,

$f(x)\rightarrow -\infty$ as $x\rightarrow -\infty$

And, $f(x)\rightarrow -\infty$ as $x\rightarrow +\infty$