Find the Critical Points f(x)=x^3+6x^2-15x

Problem

ƒ(x)=x3+6*x2−15*x

Solution

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

d(ƒ(x))/d(x)=3*x2+12*x−15

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

3*x2+12*x−15=0

3. Factor out the greatest common factor to simplify the quadratic equation.

3*(x2+4*x−5)=0

4. Factor the quadratic expression inside the parentheses by finding two numbers that multiply to −5 and add to 4

3*(x+5)*(x−1)=0

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

x+5=0⇒x=−5

x−1=0⇒x=1

Final Answer

x=−5,1

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