Maximum and Minimum Values Analysis

We are asked to find the maximum and minimum values of the expression \(5x + 7y\) subject to the constraint \(|x| + |y| \le 1\).

The constraint \(|x| + |y| \le 1\) defines a region in the xy-plane. This region is bounded by the lines where \(|x| + |y| = 1\).

Let's look at the boundary \(|x| + |y| = 1\) in each quadrant:

• In the first quadrant (\(x \ge 0, y \ge 0\)): \(x + y = 1\).
• In the second quadrant (\(x \le 0, y \ge 0\)): \(-x + y = 1\).
• In the third quadrant (\(x \le 0, y \le 0\)): \(-x - y = 1\).
• In the fourth quadrant (\(x \ge 0, y \le 0\)): \(x - y = 1\).

These lines form a diamond shape (a square rotated by 45 degrees) centered at the origin. The vertices of this region are the points where these lines intersect. These vertices are:

• (1, 0)
• (-1, 0)
• (0, 1)
• (0, -1)

The region defined by \(|x| + |y| \le 1\) includes all points inside and on the boundary of this diamond shape.

The expression we want to find the maximum and minimum values for is \(5x + 7y\). This is a linear function of \(x\) and \(y\). For a linear function defined over a closed and bounded convex region (like our diamond shape), the maximum and minimum values occur at the vertices of the region.

Therefore, we need to evaluate the expression \(5x + 7y\) at each of the vertices:

• At vertex (1, 0): \(5(1) + 7(0) = 5 + 0 = 5\).
• At vertex (-1, 0): \(5(-1) + 7(0) = -5 + 0 = -5\).
• At vertex (0, 1): \(5(0) + 7(1) = 0 + 7 = 7\).
• At vertex (0, -1): \(5(0) + 7(-1) = 0 - 7 = -7\).

The values of the expression at the vertices are 5, -5, 7, and -7.

Finding the Maximum and Minimum Values

Comparing the values obtained at the vertices:

• The maximum value is the largest among 5, -5, 7, -7, which is 7.
• The minimum value is the smallest among 5, -5, 7, -7, which is -7.

So, the maximum value of \(5x + 7y\) is 7, and the minimum value is -7.

The maximum and minimum values are 7 and -7.