For simplicity let $w_1 = 1, \; w_2 = -1$, then draw the contours of both functions:

Clearly maximum and minimum are reached at the boundary points $(0, B)$ and $(B, 0)$ respectively.

Note that if change the roles of $f_1, \;f_2$, then things will be a little different:

In this case $f_2$ is bounded below by 0 at the origin, but not bounded above.

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