Math Problem Statement

Given that -4 ≤ x ≤ -1 and 2 ≤ y ≤ 5, find (a) the greatest possible value of x + y, (b) the greatest possible value of (x - y)^2, (c) the smallest possible value of x / y.

Solution

Let’s analyze and solve each part of this question.

We are given: 4x1and2y5-4 \leq x \leq -1 \quad \text{and} \quad 2 \leq y \leq 5 We need to determine:

Part (a)

The greatest possible value of x+yx + y.

To maximize x+yx + y, we should choose the largest values of xx and yy within their respective ranges.

  • Largest x=1x = -1
  • Largest y=5y = 5

Thus: x+y=1+5=4x + y = -1 + 5 = 4 The greatest possible value of x+yx + y is 4.

Part (b)

The greatest possible value of (xy)2(x - y)^2.

To maximize (xy)2(x - y)^2, we need to maximize the absolute value of xyx - y (since squaring makes the value positive).

Let's evaluate xyx - y for the extreme values:

  1. If x=4x = -4 and y=5y = 5: xy=45=9(xy)2=(9)2=81x - y = -4 - 5 = -9 \quad \Rightarrow \quad (x - y)^2 = (-9)^2 = 81

  2. If x=4x = -4 and y=2y = 2: xy=42=6(xy)2=(6)2=36x - y = -4 - 2 = -6 \quad \Rightarrow \quad (x - y)^2 = (-6)^2 = 36

  3. If x=1x = -1 and y=5y = 5: xy=15=6(xy)2=(6)2=36x - y = -1 - 5 = -6 \quad \Rightarrow \quad (x - y)^2 = (-6)^2 = 36

  4. If x=1x = -1 and y=2y = 2: xy=12=3(xy)2=(3)2=9x - y = -1 - 2 = -3 \quad \Rightarrow \quad (x - y)^2 = (-3)^2 = 9

The greatest value of (xy)2(x - y)^2 is 81.

Part (c)

The smallest possible value of xy\frac{x}{y}.

To minimize xy\frac{x}{y}, we want xx to be as small as possible and yy to be as large as possible (a large denominator with a small numerator minimizes the fraction).

  • Smallest x=4x = -4
  • Largest y=5y = 5

Thus: xy=45=0.8\frac{x}{y} = \frac{-4}{5} = -0.8

The smallest possible value of xy\frac{x}{y} is -0.8.

Summary of Results:

  • (a) The greatest possible value of x+yx + y is 4.
  • (b) The greatest possible value of (xy)2(x - y)^2 is 81.
  • (c) The smallest possible value of xy\frac{x}{y} is -0.8.

Would you like further details on any of these calculations?


Here are 5 related questions to expand on this topic:

  1. How would the answers change if the range for xx and yy were modified?
  2. What is the smallest possible value of x+yx + y?
  3. Can we find the smallest possible value of (xy)2(x - y)^2?
  4. How would the greatest possible value of xy\frac{x}{y} be determined?
  5. What if we needed to find the greatest possible value of xyx \cdot y?

Tip: When maximizing or minimizing expressions with multiple variables in a given range, start by testing boundary values. This approach often provides the extreme values needed.

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Math Problem Analysis

Mathematical Concepts

Algebra
Optimization
Inequalities

Formulas

x + y
(x - y)^2
x / y

Theorems

Optimization through boundary values

Suitable Grade Level

Grades 10-12