What is the smallest value y can have? can anyone show how to solve?
I assume you are asking for the minimum value on a regions bounded by some lines buy and make no sense. If then x must be " . Did you mean " and ?
If so then the minimum value must occur:
1) in the interior of the set where the gradient is 0 (or undefined) or
2) on the boundary of the set.
which is never 0 because of the " " term. On the line x= 0, the y axis, the function is which has its minimum at y= 0. On the line y= 0, the x-axis, the function is which has its minimum at x= 0. On the line x+ y= 1, x= 1- y so the function is . The derivative of that is and is 0 at . Then .
Of course, we also need to consider the vertices (1, 0) and (0, 1). That is, the minimum value must occur at one of (0, 0), (1, 0), (0, 1), or . Evaluate at each of those points to decide where it is smallest.