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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.

User Leoh
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1 Answer

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Answer:


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)

User Rmic
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