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## Homework Statement

I am trying to minimize the function ##f(a) = (1+4a^2)^3 \left( \frac{1}{4a^2} \right)^2##. Here we are given that ##a>0##

## Homework Equations

Definition of a minimum of a function

## The Attempt at a Solution

Now the derivative here will be ugly and equating it to zero and solving it will be messy. So I did a substitution, ##\alpha = 4a^2##. With this the function becomes ##f(\alpha) = \frac{(1+\alpha)^3}{\alpha^2}##. The derivative is easier to calculate $$f'(\alpha) = \frac{(1+\alpha)^2}{\alpha^2}\left[ 3 - \frac{2(1+\alpha)}{\alpha} \right] $$ Now since ##a>0##, we have ##\alpha >0## and when we equate ##f'(\alpha)## to zero, the equation just becomes $$\left[ 3 - \frac{2(1+\alpha)}{\alpha} \right] = 0$$ which is much easier to solve. ##\alpha = 2##. And using the relationship between ##a## and ##\alpha##, we can see that the function has a critical point when ##a = \frac{1}{\sqrt{2}}##. With either first derivative test or the second derivative test, we can confirm that this is where the function has a local minimum. Now is it ok to do change of variable like I have done here in maxima minima problems. How do I justify this ?

Thanks

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