Find the Inflection Points f(x)=3x^3-36x-9

Problem

ƒ(x)=3*x3−36*x−9

Solution

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

d(ƒ(x))/d(x)=9*x2−36

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

d2(ƒ(x))/(d(x)2)=18*x

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

18*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 18*x is negative for x<0 and positive for x>0 an inflection point exists at x=0

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

ƒ(0)=3*(0)3−36*(0)−9

ƒ(0)=−9

Final Answer

Inflection Point=(0,−9)

Want more problems? Check here!