Questions: f(x)=4 x(x-3)^2(x+6)^3

Transcript text: $f(x)=4 x(x-3)^{2}(x+6)^{3}$

Solution

Solution Steps

To analyze the function $ f(x) = 4x(x-3)^2(x+6)^3 $, we can follow these steps:

Identify the roots: Set $ f(x) = 0 $ and solve for $ x $.
Determine the multiplicity of each root: The exponents in the factors indicate the multiplicity.
Analyze the behavior around each root: Use the multiplicity to determine if the graph touches or crosses the x-axis at each root.

Step 1: Identify the Roots

To find the roots of the function $ f(x) = 4x(x-3)^2(x+6)^3 $, we set $ f(x) = 0 $. The roots are determined to be: \[ x = -6, \quad x = 0, \quad x = 3 \]

Step 2: Determine the Multiplicity of Each Root

The multiplicity of each root is derived from the factors of the function:

For $ x = -6 $, the factor $ (x + 6)^3 $ indicates a multiplicity of $ 3 $.
For $ x = 0 $, the factor $ x $ indicates a multiplicity of $ 1 $.
For $ x = 3 $, the factor $ (x - 3)^2 $ indicates a multiplicity of $ 2 $.

Thus, the multiplicities are: \[ \text{Multiplicity of } x = -6: 3, \quad \text{Multiplicity of } x = 0: 1, \quad \text{Multiplicity of } x = 3: 2 \]

Step 3: Analyze the Behavior Around Each Root

At $ x = -6 $ (multiplicity $ 3 $): The graph crosses the x-axis.
At $ x = 0 $ (multiplicity $ 1 $): The graph crosses the x-axis.
At $ x = 3 $ (multiplicity $ 2 $): The graph touches the x-axis and does not cross.