Find the Critical Points y=x^3-3x^2-72x

Problem

y=x3−3*x2−72*x

Solution

1. Find the derivative of the function with respect to x to determine the rate of change.

d(y)/d(x)=3*x2−6*x−72

2. Set the derivative to zero because critical points occur where the slope of the tangent line is zero or undefined.

3*x2−6*x−72=0

3. Factor out the greatest common factor to simplify the quadratic equation.

3*(x2−2*x−24)=0

4. Factor the quadratic expression inside the parentheses by finding two numbers that multiply to −24 and add to −2

3*(x−6)*(x+4)=0

5. Solve for x by setting each linear factor equal to zero.

x−6=0⇒x=6

x+4=0⇒x=−4

Final Answer

x=6,−4

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