I have solved the question, I just want to know whether my reasoning for the fourth option is correct or not.
As per the 4th option, the minimum value of $f$ is attained only at finite no. of points.
So can i say, the 4th option is false since there are infinite no. of points that gives the minimum value of $f$ i.e all the numbers in $ [-1,1] $ gives minimum value. And there are infinite points between them.
Is my reasoning correct?
As has already been stated in the comments, your result and reasoning are correct.