Answer:
First, lets find the zeros.
-x4 + x3 + 7x2 - x - 6 = 0
Using synthetic division, the left side factors to
(x - 3)(-x3 - 2x2 + x + 2) = 0
Then by group factoring the second factor,
(x - 3)[-x3 + x - 2x2 + 2] = 0
(x - 3)[-x(x2 - 1) - 2(x2 - 1)] = 0
(x - 3)(x2 - 1)(-x - 2) = 0
(x - 3)(x - 1)(x + 1)(-x - 2) = 0
Our zeros are then
x = -2
x = -1
x = 1
x = 3
The company will break even when 1 tree is cut down or 3 trees are cut down.
Since we have a negative 4th degree function, the graph will start off increasing, and then end off decreasing. This is a symmetrical function. Using this fact, we know can tell the shape the graph. It will have 2 maximums and 1 minimum. There is a maximum in the interval (-2, -1), a minimum in the interval (-1, 1), and another maximum in the interval (1, 3). This is in the general sense of the graph.
However, the number of trees must be positive. So the company makes a minimum profit when they don't cut down any trees. The company makes a maximum profit when 2 trees are cut down. When no trees are cut down, the company loses profit by 6.
As for the end behaviors, x cannot be negative because you cannot cut down negative number of trees. Therefore, the graph starts at x=0.
The profits start to increase when at most 2 trees are cut down.
After 2 trees are cut down, the profits decrease.
To get a better visual, you can draw the graph on a graphing calculator.
Step-by-step explanation:
