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

Problem

ƒ(x)=2*x3+x2+2*x

Solution

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

d(ƒ(x))/d(x)=6*x2+2*x+2

2. Set the derivative equal to zero to find the critical points, as critical points occur where ƒ(x)′=0 or is undefined.

6*x2+2*x+2=0

3. Simplify the equation by dividing all terms by the greatest common factor, which is 2

3*x2+x+1=0

4. Evaluate the discriminant D=b2−4*a*c to determine if there are real solutions for x

D=1−4*(3)*(1)

D=1−12

D=−11

5. Conclude that since the discriminant is negative (D<0, there are no real solutions to the equation ƒ(x)′=0

Final Answer

No real critical points

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