What Is the Maximum Value of \(a^2+ab+2b^2\) Given \(a^2-ab+2b^2=8\)?

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In summary, the purpose of finding the maximum of a²+ab+2b² is to determine the highest possible value that the expression can have, which can be useful in various mathematical and scientific applications. To find the maximum, completing the square can be used, with the maximum value occurring when a+b is equal to zero. The maximum of a²+ab+2b² cannot be negative, as the expression contains only squared terms. The values of a and b can greatly affect the maximum, with larger values resulting in a larger maximum value. Real-world applications of finding the maximum include economics and engineering, making it a fundamental tool in various fields of science and technology.
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anemone
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$a$ and $b$ are positive real numbers such that $a^2-ab+2b^2=8$.

Find the maximum value of $a^2+ab+2b^2$.
 
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Here:

$\frac{4}{7}(4+\sqrt2)^2$
 
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conscipost said:
Here:

$\frac{4}{7}(4+\sqrt2)^2$

Your answer is correct, conscipost! (Yes) Well done!

But I'd appreciate it if you show your solution (but not merely the final answer) so we know what approach you used, sounds good to you?
 

FAQ: What Is the Maximum Value of \(a^2+ab+2b^2\) Given \(a^2-ab+2b^2=8\)?

1. What is the purpose of finding the maximum of a²+ab+2b²?

The purpose of finding the maximum of a²+ab+2b² is to determine the highest possible value that the expression can have. This can be useful in various mathematical and scientific applications, such as optimizing functions and solving equations.

2. How do you find the maximum of a²+ab+2b²?

To find the maximum of a²+ab+2b², we can use the method of completing the square. This involves rewriting the expression in the form (a+b)²+c, where c is a constant. The maximum value will occur when a+b is equal to zero, which can be solved using basic algebraic techniques.

3. Can the maximum of a²+ab+2b² be negative?

No, the maximum of a²+ab+2b² cannot be negative. This is because the expression contains only squared terms, which are always positive. Therefore, the maximum value of the expression will also be positive.

4. How does the value of a and b affect the maximum of a²+ab+2b²?

The values of a and b can greatly affect the maximum of a²+ab+2b². For example, if a is positive and b is negative, the maximum value will be different than if both a and b are positive. In general, the larger the values of a and b, the larger the maximum value of the expression will be.

5. Are there any real-world applications of finding the maximum of a²+ab+2b²?

Yes, there are many real-world applications of finding the maximum of a²+ab+2b². For example, in economics, this type of problem can be used to maximize profits or minimize costs. In engineering, it can be used to optimize the design of structures or systems. Overall, finding the maximum of a mathematical expression is a fundamental tool in various fields of science and technology.

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