how do you find critical numbers??

Question

Find the critical number of the function-
f(x)= x^3 + x^2 + x

could you explain it step by step

Answer 1

x(x^2+x+1)one of the critical numbers is 0 and the other two are imaginary.

Answer 2

The critical numbers of a function determine where the slope of the function is equal to 0. These points will usually be at a local minimum or maximum of the function.
f(x)= x^3 + x^2 + x
f'(x) = 3x^2 + 2x + 1 = 0
x = (-2 ± √(4 - 12))/6
x = -2/3(1 ± i√2)
The critical points of this function are complex, so there will be no 0-slope in the "real" x-y plane.

Answer 3

"critical numbers" has different meaning to different people, but it might include the 0's, the local max's and mins, and points of inflection.

0's are 0 and (-1 +- sqrt(-3))/2

Where are the maxs and mins? Where the derivative is 0:
f'(x) = 3x^2 + 2x + 1 = 0 x = -1/3 +- sqrt(4 - 12)/6
= -1/3 +- isqrt(2)/3
so there are no max's and mins.

Where is the point of inflection? Where the 2nd derivative is 0:

f''(x) = 6x + 2 = 0, x = -1/3