`f(x) = 2x^3 + x^2 + 2x` Find the critical numbers of the function

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to evaluate the critical numbers of the function, hence, you need to evaluate the soutions to the first derivative, such that:

`f'(x) = 0`

`f'(x) = (2x^3 + x^2 + 2x)`

`f'(x) = 6x^2 + 2x + 2`

You need to solve for x the equation f'(x) = 0:

`6x^2 + 2x + 2= 0`

You need to divide by 2:

`3x^2 + x + 1 = 0`

Using quadratic formula, yields:

`x_(1,2) = (-1+-sqrt(1 - 4*3))/(2*3) => x_(1,2) = (-1+-sqrt(-11))/6`

Notice that `Delta = -11 < 0` , hence, `x_(1,2) !in R`

Hence, evaluating the critical values of the function yields that there are no real values for f'(x) = 0.