Find the Critical Points f(x)=2x^3+9x^2-24x

Problem

ƒ(x)=2*x3+9*x2−24*x

Solution

1. Find the derivative of the function ƒ(x) with respect to x using the power rule.

d(ƒ(x))/d(x)=6*x2+18*x−24

2. Set the derivative equal to zero to find the values of x where the slope of the tangent line is zero.

6*x2+18*x−24=0

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

6*(x2+3*x−4)=0

4. Factor the quadratic expression inside the parentheses by finding two numbers that multiply to −4 and add to 3

6*(x+4)*(x−1)=0

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

x+4=0⇒x=−4

x−1=0⇒x=1

Final Answer

x=−4,1

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