Chapter 6.4, Problem 4BCYP

To determine

To find:the turning point and end behavior of the graph.

Answer to Problem 4BCYP

Turning point here is $\left(2.121,1339.628\right)$ andthe end behavior is down and down.

Explanation of Solution

Given:

  $\text{f}\left(\text{x}\right)\text{=-0}{\text{.17x}}^{\text{4}}\text{+6}{\text{.29x}}^{\text{3}}\text{-77}{\text{.65x}}^{\text{2}}\text{+251x+1100}$ .

Concept used:

Turning point: a point in curve where the slope of graph changes from negative to positive or positive to negative.

End behavior refers to the direction of the two end points of the graph.

Case 1. If the highest degree of the equation is even the two end points will in same direction.

Case2. If the highest degree of the equation is odd the two end points will be in opposite direction.

If the highest degree of the equation is even and sign of the coefficient of the highest degree is positive then the curve will open upward the direction of the two end points will be up and up.

Similarly,

If the highest degree of the equation is even and sign of the coefficient of the highest degree is negative then the curve will open downward the direction of the two end points will be down and down.

Calculation:

  $\text{f}\left(\text{x}\right)\text{=-0}{\text{.17x}}^{\text{4}}\text{+6}{\text{.29x}}^{\text{3}}\text{-77}{\text{.65x}}^{\text{2}}\text{+251x+1100}$ .

The graph of given function $\text{f}\left(\text{x}\right)\text{=-0}{\text{.17x}}^{\text{4}}\text{+6}{\text{.29x}}^{\text{3}}\text{-77}{\text{.65x}}^{\text{2}}\text{+251x+1100}$ is:

  

Hereturning point here is $\left(2.121,1339.628\right)$ and the curve is open down ward since the sign of the highest degree coefficient is negative and highest degree itself is even so, the end behavior is down and down.