Find the Inflection Points f(x)=x^3+6x^2+9x

Problem

ƒ(x)=x3+6*x2+9*x

Solution

1. Find the first derivative by applying the power rule to each term of the function.

d(ƒ(x))/d(x)=3*x2+12*x+9

2. Find the second derivative by differentiating the first derivative with respect to x

d2(ƒ(x))/(d(x)2)=6*x+12

3. Set the second derivative to zero to find potential inflection points where the concavity might change.

6*x+12=0

4. Solve for x by isolating the variable.

6*x=−12

x=−2

5. Verify the inflection point by checking the sign of the second derivative on either side of x=−2 Since ƒ(x)″ is a linear function with a positive slope, it changes from negative to positive at x=−2 confirming a change in concavity.

6. Calculate the y-coordinate by substituting x=−2 back into the original function ƒ(x)

ƒ*(−2)=(−2)3+6*(−2)2+9*(−2)

ƒ*(−2)=−8+24−18

ƒ*(−2)=−2

Final Answer

Inflection Point=(−2,−2)

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