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

Problem

ƒ(x)=x3−3*x

Solution

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

d(ƒ(x))/d(x)=3*x2−3

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

d2(ƒ(x))/(d(x)2)=6*x

3. Identify potential inflection points by setting the second derivative equal to zero and solving for x

6*x=0

x=0

4. Test for concavity changes by checking the sign of the second derivative on intervals around x=0
   For x<0 ƒ(x)″<0 (concave down).
   For x>0 ƒ(x)″>0 (concave up).

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

ƒ(0)=(0)3−3*(0)

ƒ(0)=0

Final Answer

Inflection Point=(0,0)

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