Find the Inflection Points f(x)=x^3+42x^2

Problem

ƒ(x)=x3+42*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+84*x

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

d2(ƒ(x))/(d(x)2)=6*x+84

3. Set the second derivative to zero to find potential inflection points.

6*x+84=0

4. Solve for x by subtracting 84 from both sides and then dividing by 6

6*x=−84

x=−14

5. Verify the concavity change by checking the sign of ƒ(x)″ on either side of x=−14 Since ƒ(x)″ is a linear function with a positive slope, it changes from negative to positive at x=−14 confirming an inflection point.

6. Calculate the y-coordinate by substituting x=−14 back into the original function ƒ(x)

ƒ*(−14)=(−14)3+42*(−14)2

ƒ*(−14)=−2744+42*(196)

ƒ*(−14)=−2744+8232

ƒ*(−14)=5488

Final Answer

Inflection Point=(−14,5488)

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