Find the Inflection Points y=x^3-8x^2-12x+9

Problem

y=x3−8*x2−12*x+9

Solution

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

d(y)/d(x)=3*x2−16*x−12

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

d2(y)/(d(x)2)=6*x−16

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

6*x−16=0

4. Solve for x by isolating the variable.

6*x=16

x=16/6

x=8/3

5. Find the y-coordinate by substituting the x value back into the original function y=x3−8*x2−12*x+9

y=(8/3)3−8*(8/3)2−12*(8/3)+9

y=512/27−512/9−32+9

y=512/27−1536/27−23

y=−1024/27−621/27

y=−1645/27

6. Verify the inflection point by checking if the second derivative changes sign around x=8/3 Since 6*x−16 is a linear function with a non-zero slope, the sign changes, confirming an inflection point.

Final Answer

Inflection Point=(8/3,−1645/27)

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