Find the Critical Points f(x)=3x^4-4x^3-12x^2+24

Problem

ƒ(x)=3*x^4−4*x^3−12*x^2+24

Solution

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

d(ƒ(x))/d(x)=12*x^3−12*x^2−24*x

2. Set the derivative to zero to find the critical points where the slope of the tangent line is horizontal.

12*x^3−12*x^2−24*x=0

3. Factor out the greatest common factor, which is 12*x

12*x*(x^2−x−2)=0

4. Factor the quadratic expression inside the parentheses.

12*x*(x−2)*(x+1)=0

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

12*x=0⇒x=0

x−2=0⇒x=2

x+1=0⇒x=−1

Final Answer

x=−1,0,2

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