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

Problem

ƒ(x)=3*x−x3

Solution

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

d(ƒ(x))/d(x)=3−3*x2

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

3−3*x2=0

3. Factor out the common constant to simplify the equation.

3*(1−x2)=0

4. Solve for x by factoring the difference of squares or isolating the variable.

1−x2=0

x2=1

x=±1

5. Identify the critical points as the xvalues where the derivative is zero or undefined (the derivative is defined for all real numbers in this case).

x=1,x=−1

Final Answer

x=1,−1

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