Find the Inflection Points y=x^3-6x^2-36x

Problem

y=x3−6*x2−36*x

Solution

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

d(y)/d(x)=3*x2−12*x−36

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

d2(y)/(d(x)2)=6*x−12

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

6*x−12=0

4. Solve for x by isolating the variable.

6*x=12

x=2

5. Verify the inflection point by checking if the second derivative changes sign around x=2 For x<2 d2(y)/(d(x)2)<0 (concave down), and for x>2 d2(y)/(d(x)2)>0 (concave up).

6. Calculate the y-coordinate by substituting x=2 back into the original function.

y=(2)3−6*(2)2−36*(2)

y=8−24−72

y=−88

Final Answer

Inflection Point=(2,−88)

Want more problems? Check here!