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?