Find the Critical Points x^3-27x

Problem

ƒ(x)=x3−27*x

Solution

1. Define the function and its domain. Let ƒ(x)=x3−27*x Since this is a polynomial, it is defined and differentiable for all real numbers.

2. Find the derivative of the function with respect to x using the power rule.

(d(x3)−27*x)/d(x)=3*x2−27

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

3*x2−27=0

4. Solve for x by first factoring out the greatest common factor.

3*(x2−9)=0

5. Factor the difference of squares to isolate the roots.

3*(x−3)*(x+3)=0

6. Identify the critical points by solving the resulting linear equations.

x−3=0⇒x=3

x+3=0⇒x=−3

Final Answer

x=3,−3

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