Any interior solution for $u(x,y) = min\left \{ x,y \right \}^{2} + max\left \{ x,y \right \}$? Will all the solutions be in the corner or will the cusp in the middle give us any interior solution? This is by the intersection of the budget line.

I am getting this type of a shape:


But I am not sure if there will be interior solution. If there is we can't use differentiation because of the cusp. I think that there will only be corner solutions but it depends on the shape of the indifference curve.
Because the two segments of the IC bend away from the origin, the only possible solutions are at the corners or at $x=y$.

Write $p_x$ and $p_y$ for the prices, and $M$ for the budget.

Suppose we try to set up a corner solution in which only one of the two goods is consumed. Since utility is symmetric, it makes sense to spend all of your money on whichever good is cheaper. Suppose this is good $x$. We then have $x=\frac{M}{p_x}&gt;0=y$ and $u=x$.

The best $x=y$ bundle that can be afforded is such that $x p_x+x p_y=M\iff x=y=\frac{M}{p_x+p_y}$. Utility is then
$$\frac{M^2}{(p_x+p_y)^2}+\frac{M}{p_x+p_y}.$$

You should easily be able to compute a condition on $M$ such that the bundle with $x=y$ is optimal: $$M&gt;\frac{p_y(p_x+p_y)}{p_x}.$$

You can see this dependence on M if we look at a contour plot of the indifference curves: