Find the Critical Points f(x)=3x^5-20x^3

Problem

ƒ(x)=3*x5−20*x3

Solution

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

d(ƒ(x))/d(x)=15*x4−60*x2

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

15*x4−60*x2=0

3. Factor the expression by taking out the greatest common factor, which is 15*x2

15*x2*(x2−4)=0

4. Factor the difference of squares inside the parentheses.

15*x2*(x−2)*(x+2)=0

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

15*x2=0⇒x=0

x−2=0⇒x=2

x+2=0⇒x=−2

Final Answer

x={−2,0,2}

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