Find the Inflection Points f(x)=2x^3+3x^2-12x

Problem

ƒ(x)=2*x3+3*x2−12*x

Solution

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

d(ƒ(x))/d(x)=6*x2+6*x−12

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

d2(ƒ(x))/(d(x)2)=12*x+6

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

12*x+6=0

4. Solve for x by isolating the variable.

12*x=−6

x=−1/2

5. Verify the inflection point by checking if the second derivative changes sign around x=−1/2 Since 12*x+6 is a linear function with a non-zero slope, the sign changes, confirming an inflection point.

6. Find the y-coordinate by substituting x=−1/2 back into the original function ƒ(x)

ƒ*(−1/2)=2*(−1/2)3+3*(−1/2)2−12*(−1/2)

ƒ*(−1/2)=2*(−1/8)+3*(1/4)+6

ƒ*(−1/2)=−1/4+3/4+6

ƒ*(−1/2)=1/2+6=13/2

Final Answer

Inflection Point=(−1/2,13/2)

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