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

Answer:

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

Step-by-step 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)