Sketch a graph of the polynomial function f(x) =x3− 2x2. Use it to complete the following: f is on the intervals (−∞, 0) and (0, 2).
f is on the interval (2,∞).f is on the interval (0, 4__3).
f is on the intervals (−∞, 0) and(4__3, ∞).

We have the function:

f(x) = x^3 - 2*x^2

To sketch this, we need to graph some points, and then just draw a line that passes through the points.

The graph of this equation is shown below.

Now we can complete the question.

If the graph is below the x-axis in some interval, the function is negative in that interval

If the graph is above the x-axis in some interval, the function is positive in that interval.

If the graph goes up in a interval, then the function is increasing in that interval

If the graph goes down on an interval, then the function is decreasing in that interval.

Then:

1) f is------ on the intervals (−∞, 0) and (0, 2).

Here we can see that the graph is below the x-axis in those intervals, then here we have:

f is negative on the intervals (−∞, 0) and (0, 2).

2) f is------ on the interval (2,∞)

Here the answer is positive:

f is positive on the interval  (2,∞)

3) fi is ------ on the interval (0, 4/3)

In the graph, you can see that the graph goes down in that interval, then the correct answer here is:

f is decreasing on the interval (0, 4/3)

4) f is------ on the intervals (−∞, 0) and (4/3, ∞).

In this case, we can see that the graph goes up in these intervals, then the correct answer here is:

f is increasing on the intervals (−∞, 0) and (4/3, ∞).

let x be the money she spent so that she can earn the free t-shirt.

since, she already spent $112.49

⇒

also, as per the statement: jennifer must spend $250 or more in order to qualify for a free t-shirt at the store.

then,

we have an inequality i.,e

subtract 112.49 from both sides we get;

therefore, $137. 51 more money does jennifer need to spend so that she can earn the free t-shirt