Question

g Using calculus and the SDT (then FDT if necessary), find all global and local maximum and minimums given the function f(x) = x 3 + x 2 − x + 1 where x ∈ [−2, 1 2 ]. Clearly identify critical values and show the SDT then the FDT if the SDT didn’t provide an answer and then interpret the solution.

Answer 1

`S_(1 ) (x,y) = (0.333, 0.815)` (Absolute minimum) and
`S_(2) (x,y) = (-1, 2)` (Absolute maximum)

Explanation:

The critical points are determined with the help of the First Derivative Test:

`f'(x) = 3\cdot x ^(2) +2\cdot x -1`

`3\cdot x^(2) + 2\cdot x - 1 = 0`

The critical points are:

`x_(1) \approx 0.333` and
`x_(2) \approx -1`

The Second Derivative Test offers a criterion to decide whether critical point is an absolute maximum and whether is an absolute minimum:

`f''(x) = 6\cdot x +2`

`f''(x_(1)) = 3.998` (Absolute minimum)

`f''(x_(2)) = -4` (Absolute maximum)

The critical points are:

`S_(1 ) (x,y) = (0.333, 0.815)` (Absolute minimum) and
`S_(2) (x,y) = (-1, 2)` (Absolute maximum)