Find the Inflection Points f(x)=x^3+27x^2

Problem

ƒ(x)=x3+27*x2

Solution

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

d(ƒ(x))/d(x)=3*x2+54*x

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

d2(ƒ(x))/(d(x)2)=6*x+54

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

6*x+54=0

4. Solve for x by isolating the variable.

6*x=−54

x=−9

5. Verify the inflection point by checking if the second derivative changes sign around x=−9 Since 6*x+54 is a linear function with a non-zero slope, the sign changes from negative to positive at x=−9

6. Calculate the y-coordinate by substituting x=−9 back into the original function ƒ(x)

ƒ*(−9)=(−9)3+27*(−9)2

ƒ*(−9)=−729+27*(81)

ƒ*(−9)=−729+2187

ƒ*(−9)=1458

Final Answer

Inflection Point=(−9,1458)

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