I will try to understand the theory about indifference curves correctly. I have given a Utility function $u(x_1,x_2)=x_1x_2^2$. Then I have to illustrate the bundles that are indifference with the bundle $(x_1,x_2)=(2,4)$.
I think I the get the line $x_2(x_1)=(\frac{u}{x_1})^{1/2}$ for the indifference curve. If we find $MRS(2,4)=-\frac{\frac{1}{2}x_1}{\frac{1}{2}x_2}=-\frac{1}{2}$. I'm not quite sure how to plot that? I have not any values of u. Can someone help me?
You have $u(x_1,x_2)=x_1x_2^2$. The indifference curves are the level sets of that function, that is, all values of $x_1,x_2$ such that $u(x_1,x_2)=K$ for some constant $K.$ In your particular problem, you want all $x_1,x_2$ such that $x_1x_2^2=2\times 4^2=32.$ don't know the name of that curve, but it should be possible to plot by hand or program.
Calculating the MRS is a good idea, because with the data from the problem you have both a point on the indifference curve and its slope at that point. Of course in general the slope is $-\frac{1}{2}\frac{x_2}{x_1}$ so that can give you an idea of how the slope varies.
Also note that this is just plotting out indifference curves, so prices have not entered the questions yet, so we don't have to worry about them for this question, which is just about preferences.