Find the Critical Points f(x)=3x^4+16x^3+24x^2+32

Problem

ƒ(x)=3*x4+16*x3+24*x2+32

Solution

1. Find the derivative of the function ƒ(x) using the power rule for each term.

d(ƒ(x))/d(x)=12*x3+48*x2+48*x

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

12*x3+48*x2+48*x=0

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

12*x*(x2+4*x+4)=0

4. Factor the quadratic expression inside the parentheses, noting that it is a perfect square trinomial.

12*x*(x+2)2=0

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

12*x=0⇒x=0

(x+2)2=0⇒x=−2

Final Answer

x=0,−2

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