An utility function that exhibits increasing MRSx,y is U(x,y) = Ax2 + By2.
MRSx,y = MUx/MUy where MUx is the marginal utility of good x and MUy is the marginal utility of good y.
For U(x,y) = Ax2 + By2, MUx = 2Ax, MUy = 2By.
So, MRSx,y = 2Ax/2By = Ax/By, which increases as the consumer increases consumption of x and reduces the consumption of y along an indifference curve. Thus, this utility function exhibits increasing marginal rate of substitution. The indifference curves corresponding to this utility function are concave to the origin.