Question

whether the minima of some function f(x) and minimum of the same function f(x)/2 are same?

Answer 1

it depends upon interval

Answer 2

if \[f(x_\min) = \min\]then \[k \times f(x_\min) = k \times \min\]for every\[k \neq 0\]The same for maxima.

Answer 3

Depends of the minum value.

If min(f(x)) is non zero at point p then min(f(x)/2) is not the same at point p, but if min(f(x)) is 0 at point p then min(f(x)/2) is also 0 at point p and thus the minimums are the same.

Simple example is y=x^2+2 which has minimum at x=0 and the value of the minmum is 2. If you take half of that function it still has the minimum at x=0, but the value of the minimum is 1.

On the other hand y=x^2 has minimum at x=0 and the value of that minimum is 0. If you take half of that function it still has minimum at point 0 and it's value 0.