the average amount of oranges in pounds eaten per person each year in the 6 can be modeled byunited states from 1991 to 1996 by

0.298x^3-2.73x^2+7.05x+8.45

where x is the number of years since 1991. Graph the function and identify any turning points on the interval 0<x<5. What real life meaning do these points have?

If you do the graph, you'll see that it turns downward at t = 1.85, then turns back up at t=4.25

So, consumption of oranges increased until about November of 1992, then decreased through March of 1995, when it started to increase again.

Don't know what happened during those three years; drought? freeze? disease? check elsewhere for that info.

To graph the function, we will plot points on a coordinate system using the given equation.

The equation is: f(x) = 0.298x^3 - 2.73x^2 + 7.05x + 8.45

First, let's find the turning points by taking the derivative of the function and setting it equal to zero.

f'(x) = 0.894x^2 - 5.46x + 7.05

Setting f'(x) = 0, we have:

0.894x^2 - 5.46x + 7.05 = 0

Solving this quadratic equation, we find two possible values for x:

x ≈ 3.36 and x ≈ 2.06

These are the x-coordinates of the potential turning points. Now, let's graph the function and identify these points:

To graph the function, we will plot several points for various x-values on the interval 0 < x < 5. Here are some values:

x = 1: f(1) ≈ 13.093
x = 2: f(2) ≈ 12.516
x = 3: f(3) ≈ 16.537
x = 4: f(4) ≈ 26.640

Plotting these points on a graph will help us identify any turning points.

In the graph, we observe that there is a maximum turning point at approximately (3.36, 16.537) and a minimum turning point at approximately (2.06, 12.516).

The real-life meaning of these turning points could be interpreted as follows:

The maximum turning point represents the peak year in terms of the average amount of oranges eaten per person, indicating a high consumption of oranges during that period. This could be due to various factors such as increased awareness of the health benefits of oranges, availability, and affordability.

The minimum turning point represents a dip in orange consumption, indicating a lower average amount of oranges eaten per person during that year. Factors such as changes in dietary preferences, availability, or affordability of oranges could contribute to this decrease.

It's important to note that this interpretation is based solely on the assumption that the given function accurately models the average amount of oranges eaten per person each year. Real-life data and additional context are necessary to validate these interpretations.

To graph the function and find the turning points, you need to plot the function on a graph and determine where the curve changes direction.

To do this, you can start by creating a table of values for x and calculating the corresponding y-values using the given function. Here's how you can do it:

1. Choose several values for x within the interval 0 < x < 5. For example, you can use x = 0, 1, 2, 3, 4, and 5.
2. Plug in each value of x into the function f(x) = 0.298x^3 - 2.73x^2 + 7.05x + 8.45 and calculate the corresponding y-value.

The table of values would look like this:

x | f(x)
---------
0 | 8.45
1 | 13.078
2 | 11.072
3 | 7.218
4 | 6.828
5 | 11.758

Now, you can plot these points on a graph with x on the horizontal axis and f(x) on the vertical axis. Connect the points with a smooth curve.

To identify the turning points, look for points on the graph where the curve changes from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). These are the points where the slope of the curve changes sign.

In this case, the function is a cubic polynomial, so it can have either two or zero turning points within the interval 0 < x < 5. To determine if there are any turning points, you can find its derivative and analyze its sign changes.

The derivative of the function f(x) = 0.298x^3 - 2.73x^2 + 7.05x + 8.45 is:
f'(x) = 0.894x^2 - 5.46x + 7.05

To find the turning points, solve the equation f'(x) = 0. Any real solutions to this equation correspond to potential turning points. Use algebraic methods like factoring, completing the square, or the quadratic formula to find the solutions.

Once you have the solutions, substitute them back into the original function f(x) and calculate the corresponding y-values. These points will have a real-life meaning based on the given context of the average amount of oranges eaten per person each year in the United States from 1991 to 1996. The turning points represent significant changes or shifts in the trend of orange consumption during this period.