Find the Inflection Points f(x)=4x^3-12x

Problem

ƒ(x)=4*x3−12*x

Solution

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

d(ƒ(x))/d(x)=12*x2−12

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

d2(ƒ(x))/(d(x)2)=24*x

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

24*x=0

4. Solve for x to determine the coordinate where the concavity might change.

x=0

5. Verify the inflection point by checking if the second derivative changes sign around x=0 Since 24*x<0 for x<0 and 24*x>0 for x>0 the concavity changes.

6. Calculate the y-coordinate by substituting x=0 back into the original function ƒ(x)

ƒ(0)=4*(0)3−12*(0)

ƒ(0)=0

Final Answer

(0,0)

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