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Que:- Consider the function f(x) = $2x^3 - 3x^2$ in the domain [-1, 2], the "global minimum" value of f(x) is _______?
(A) -5  (B)-1  (C)4   (D) 0

Solution:
f(x) = $2x^3 - 3x^2$
f'(x) = $6x^2 -6x$ =0, x=0,1 ( critical points)

f"(x) = 12x -6
f"(0) = 12*0 - 6 < 0  [f(x) attains local maximum at x=0]
f"(1) = 12*1 - 6 = 6 >0 [f(x) attains local minimum at x=1]

Consider extreme points also as closed intervals are given:

f(1) = -1 ( local minimum)
f(-1) = -5
f(2) = 4

Min(-1, -5, 4) = -5( global minimum)

In this question "global minium" is asked hence answer is -5.
My question is, if "local minimum" is asked instead of "global minimum" then what will be the answer -5 or -1, as closed intervals are given, so in case of local minimum also we should consider the extreme points, right?

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