Find the Critical Points f(x)=x^4-2x^2

Problem

ƒ(x)=x4−2*x2

Solution

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

d(ƒ(x))/d(x)=4*x3−4*x

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

4*x3−4*x=0

3. Factor the expression by taking out the greatest common factor, which is 4*x

4*x*(x2−1)=0

4. Factor the difference of squares inside the parentheses to fully decompose the polynomial.

4*x*(x−1)*(x+1)=0

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

4*x=0⇒x=0

x−1=0⇒x=1

x+1=0⇒x=−1

Final Answer

x=−1,0,1

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