I have just attached another figure.Needless to say, mavropnevma's solution is perfect.

P.S.I remember that the similar problem original problem has been posed in 1970's in Tokyo University entrance exam/Science.

According to my memory, let

region

be the maximum area of any triangle which is involved in the

to find the extrema of

. in original problem, then draw the graph of

Here is the similar problem posed in Tokyo University entrance exam/Science, second round, 1978

In the

-plane, let

be the part which is correspond to

. of the parabola ,

that is to say,