Find the Inflection Points y=x^3-3x^2-105x

Problem

y=x3−3*x2−105*x

Solution

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

d(y)/d(x)=3*x2−6*x−105

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

d2(y)/(d(x)2)=6*x−6

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

6*x−6=0

4. Solve for x by isolating the variable.

6*x=6

x=1

5. Verify the inflection point by checking if the second derivative changes sign around x=1 Since 6*x−6 is a linear function with a positive slope, it changes from negative to positive at x=1

6. Calculate the y-coordinate by substituting x=1 back into the original function.

y=(1)3−3*(1)2−105*(1)

y=1−3−105

y=−107

Final Answer

(1,−107)

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