Math Problem Statement
Solution
Let’s analyze and solve each part of this question.
We are given: We need to determine:
Part (a)
The greatest possible value of .
To maximize , we should choose the largest values of and within their respective ranges.
- Largest
- Largest
Thus: The greatest possible value of is 4.
Part (b)
The greatest possible value of .
To maximize , we need to maximize the absolute value of (since squaring makes the value positive).
Let's evaluate for the extreme values:
-
If and :
-
If and :
-
If and :
-
If and :
The greatest value of is 81.
Part (c)
The smallest possible value of .
To minimize , we want to be as small as possible and to be as large as possible (a large denominator with a small numerator minimizes the fraction).
- Smallest
- Largest
Thus:
The smallest possible value of is -0.8.
Summary of Results:
- (a) The greatest possible value of is 4.
- (b) The greatest possible value of is 81.
- (c) The smallest possible value of is -0.8.
Would you like further details on any of these calculations?
Here are 5 related questions to expand on this topic:
- How would the answers change if the range for and were modified?
- What is the smallest possible value of ?
- Can we find the smallest possible value of ?
- How would the greatest possible value of be determined?
- What if we needed to find the greatest possible value of ?
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
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