Determine $m\in\mathbb{R}$ such that the solution (x,y) is unique

Question

$$x^2-y+m \le 0$$ $$y^2-x+m \le 0$$ Which value in $mathbb{R}$ does $m$ take such that the above system has a unique solution of the form $(x,y)$

Answer 1

Since graphs of $y=x^2+m$ and of $x=y^2+m$ are symmetric in relation to the graph of $y=x$,

we'll get an unique solution, when $y=x$ is a tangent line to $y=x^2+m$.

Thus, the equation $x^2-x+m=0$ has one root, which happens for $m=\frac{1}{4}$.

Answer 2

Hint. Try to draw a picture to see how the parabolas $y=x^2+m$ and $x=y^2+m$ change if you change $m$.