Problem 27 - Precalculus: Functions and Graphs

Question

Find all values of \(x\) such that \(f(x)>0\) and all \(x\) such that \(f(x)<0,\) and sketch the graph of \(f\) $$f(x)=x^{2}(x+2)(x-1)^{2}(x-2)$$

Step-by-Step Solution

Answer

\(f(x) > 0\) on \((-\infty, -2), (0, 1), (2, \infty)\); \(f(x) < 0\) on \((-2, 0), (1, 2)\).

Step 1: Identify critical points

To find where \(f(x)\) changes sign, we first find the roots of the polynomial. Factor \(f(x) = x^2(x+2)(x-1)^2(x-2)\). The critical points are at \(x = 0, -2, 1, 2\). At these points, \(f(x) = 0\).

Step 2: Determine the intervals

The critical points split the number line into intervals: \((-\infty, -2)\), \((-2, 0)\), \((0, 1)\), \((1, 2)\), and \((2, \infty)\). We will test the sign of \(f(x)\) in each interval.

Step 3: Test each interval

- Choose a test point in each interval and calculate the sign of \(f(x)\): - \((-\infty, -2)\): Test point \(x = -3\), \(f(-3) > 0\). - \((-2, 0)\): Test point \(x = -1\), \(f(-1) < 0\). - \((0, 1)\): Test point \(x = 0.5\), \(f(0.5) > 0\). - \((1, 2)\): Test point \(x = 1.5\), \(f(1.5) < 0\). - \((2, \infty)\): Test point \(x = 3\), \(f(3) > 0\).

Step 4: Determine where \(f(x) > 0\) and \(f(x) < 0\)

From the test points, we see:- \(f(x) > 0\) on the intervals \((-\infty, -2)\), \((0, 1)\), and \((2, \infty)\).- \(f(x) < 0\) on the intervals \((-2, 0)\) and \((1, 2)\).

Step 5: Sketch the graph

On a graph, plot the critical points where \(f(x) = 0\): \(-2, 0, 1,\) and \(2\). Indicate the sign of \(f(x)\) in each interval on the graph. The graph should cross the x-axis at these critical points and be above the x-axis where \(f(x) > 0\) and below where \(f(x) < 0\). Note that there is a touch at \(x=1\) where there is a double root, leading to \(f(x)\) not changing sign.

Key Concepts

Critical Points

Critical points in a polynomial function help us identify where the function crosses or touches the x-axis. For the given function \(f(x) = x^2(x+2)(x-1)^2(x-2)\), finding these points is the first step to understanding the behavior of the graph.
Simply set the polynomial equal to zero and solve for \(x\):

• The term \(x^2\) yields a critical point at \(x = 0\).
• From \(x + 2\), a critical point arises at \(x = -2\).
• The factor \((x - 1)^2\) results in a critical point at \(x = 1\), but as a double root, the graph merely touches the x-axis here.
• The term \(x - 2\) provides a critical point at \(x = 2\).

These points divide the real number line into different intervals for further analysis.

Sign of Polynomial

Determining the sign of a polynomial is crucial for understanding where the function is positive or negative. We analyze the sign by testing values in each interval created by the critical points.
Crucial for this process is to remember how odd and even multiplicities affect the sign.

• A root with an even multiplicity does not cause the graph to cross the x-axis. Instead, it touches and returns the same way.
• Conversely, an odd multiplicity facilitates a crossing of the x-axis.

For this function:

• The graph crosses at \(-2\) and \(2\), and touches at \(1\), indicating a change at those points because of the multiplicities.

Interval testing will help refine this understanding as we verify which sections are greater or less than zero.

Interval Testing

After identifying intervals from critical points, we test each to determine the sign of \(f(x)\). Choose a test point within each segment:

• For \((-\infty, -2),\) choose \(x = -3\). The computed positive value implies \(f(x) > 0\) in this region.
• For \((-2, 0),\) select \(x = -1\). The resulting negative value shows \(f(x) < 0\).
• Test \(x = 0.5\) in \((0, 1),\) yielding a positive sign, so \(f(x) > 0\) here.
• at \((1, 2)\), using \(x = 1.5\), the outcome is negative thus \(f(x) < 0\).
• Lastly, test \(x = 3\) in range \((2, \infty),\) resulting in \(f(x) > 0\).

This tells us exactly where the polynomial is above or below the x-axis.

Polynomial Graph Sketching

Sketching the graph of a polynomial becomes relatively simple once we've analyzed the critical points and their signs. Consider the critical points \(-2, 0, 1,\) and \(2\) identified earlier. Each of these points will act as a cornerstone in constructing the graph.
As we plot these:

• At \(x = 0\) and \(x = 2\), the graph crosses the x-axis.
• At \(x = -2\) and \(x = 1\), the graph approaches but doesn't cross the x-axis, merely touching it.

Using the results from interval testing, we draw the graph:

• Above the x-axis from \((-\infty, -2)\), \((0, 1)\), and \((2, \infty)\).
• Below the x-axis between \((-2, 0)\) and \((1, 2)\).

With these aspects, you can visualize a wave-like graph that respects the calculated signs and root behaviors.