How do you find the critical numbers for #f(x)= x^3 + x^2 + x# to determine the maximum and minimum?

Question

How do you find the critical numbers for #f(x)=  x^3 +  x^2  + x# to determine the maximum and minimum?

William Abdalla · Accepted Answer

It doesn't have any.

The critical values for a function are x-values where #f'(x)=0#. Differentiating the function gives: #f'(x)=3x^2+2x+1# You can either use the quadratic formula or your graphing calculator to find the zeros of this function. In the end, you'll get the fact that #f'(x)# is never equal to zero. Because of this, #f(x)# has no critical values, and no minimum or maximum.

Axel Alvarez · Answer

undefined To find the critical numbers for $ f(x) = x^3 + x^2 + x $ and determine the maximum and minimum, follow these steps:

1. Find the derivative of the function, $ f'(x) $.
2. Set $ f'(x) $ equal to zero and solve for $ x $ to find critical points.
3. Test the critical points to determine whether they correspond to a maximum or minimum or neither.

Applying these steps:

1. Find the derivative of $ f(x) $:
   $$ f'(x) = \frac{d}{dx} (x^3 + x^2 + x) $$

2. Set $ f'(x) $ equal to zero and solve for $ x $:
   $$ f'(x) = 3x^2 + 2x + 1 = 0 $$

3. Solve the quadratic equation $ 3x^2 + 2x + 1 = 0 $ for $ x $. If there are real solutions, those will be the critical points.

4. Once you have the critical points, test them using the second derivative test or by analyzing the behavior of the function around those points to determine whether they correspond to a maximum, minimum, or neither.

5. Identify the maximum and minimum points by analyzing the function values at critical points and at endpoints, if applicable.