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anemone
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If $\log_4 (a+2b)+\log_4 (a-2b)=1$, find the minimum of $|a|-|b|$.
anemone said:If $\log_4 (a+2b)+\log_4 (a-2b)=1$, find the minimum of $|a|-|b|$.
chisigma said:From the initial conditions we derive immediately...
$\displaystyle (a + 2\ b)\ (a - 2\ b) = 4 -> b = \frac{\sqrt{a^{2} - 4}}{2}\ (1)$
... so that the problem is to minimize respect to a the function...
$\displaystyle f(a) = a - \frac{\sqrt{a^{2} - 4}}{2}\ (2)$
Kind regards
$\chi$ $\sigma$
The purpose of finding the minimum of |a| - |b| is to determine the smallest possible value that can result from subtracting the absolute values of two numbers, a and b. This can be useful in various mathematical and scientific applications, such as optimization problems or finding the shortest distance between two points.
To find the minimum of |a| - |b|, you can use a number line or a graph to visualize the values and see where the lowest point is. Alternatively, you can also use algebraic methods by setting the expression equal to zero and solving for the variables a and b.
Yes, the minimum of |a| - |b| can be negative. This can happen when the absolute value of a is greater than the absolute value of b. In this case, the minimum value will be equal to the negative of the absolute value of a.
If a and b are equal, the minimum of |a| - |b| will be equal to zero. This is because subtracting the absolute values of two equal numbers will always result in zero.
Yes, there are various real-life applications for finding the minimum of |a| - |b|. For example, in physics, this can be used to find the shortest distance between two points or the minimum amount of energy required to perform a task. In economics, this can be used to find the minimum cost or price difference between two products. It can also be used in engineering and optimization problems to find the most efficient solution.