Maths: maxima - minima (global minmum vs local minimum)

Question

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?

Comments

@manu

I think local minimum will be at x=1 and if we talk about global minimum then we have to check all the values, in the sense, all critical points along with closed interval values on which function is defined

Local minimum will be x=1

the global minimum will be x=-1

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@akash yes, seems correct to me!

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yes

At x=-1, a function has global minimum value -5..