Graph f(x)=x(x-9)^2

Problem

ƒ(x)=x*(x−9)2

Solution

1. Identify the x-intercepts by setting ƒ(x)=0

x*(x−9)2=0

x=0

x=9

2. Determine the multiplicity of each root to understand the behavior at the intercepts.

At *x=0, multiplicity is *1⇒crosses the x-axis

At *x=9, multiplicity is *2⇒touches the x-axis and turns around

3. Determine the end behavior by looking at the leading term of the expanded polynomial.

ƒ(x)≈x(x2)=x3

As *x→∞,ƒ(x)→∞

As *x→−∞,ƒ(x)→−∞

4. Find the y-intercept by evaluating ƒ(0)

ƒ(0)=0*(0−9)2=0

5. Find the critical points by setting the first derivative to zero.

ƒ(x)=x*(x2−18*x+81)=x3−18*x2+81*x

d(ƒ(x))/d(x)=3*x2−36*x+81

3*(x2−12*x+27)=0

3*(x−3)*(x−9)=0

x=3,x=9

6. Evaluate the function at the critical points to find local extrema.

ƒ(3)=3*(3−9)2=3*(−6)2=108

ƒ(9)=9*(9−9)2=0

Local maximum at *(3,108)

Local minimum at *(9,0)

Final Answer

To graph ƒ(x)=x*(x−9)2 plot the x-intercepts at (0,0) and (9,0) the local maximum at (3,108) and the local minimum at (9,0) The graph starts from the bottom-left, crosses the origin, rises to (3,108) falls to touch the x-axis at (9,0) and then rises toward the top-right.

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