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From an old math exam I found the question to find the interval for when a function is decreasing(so it can be used for the Integration test). But I can't seem to figure it out.

The function in question is:

$f(x) =\dfrac{\sqrt{x}}{(x^\frac{3}{2} +2)^2}$

There is apparently an effective way to this because it was a small question with just a few points.

So can anybody show me what i am missing?

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    $\begingroup$ Have you tried taking a derivative? Wherever the derivative is negative, the function is decreasing. $\endgroup$ – BGreen Apr 4 at 22:16
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You are looking for the interval where the derivative is negative.

I will use a little trick, for comfort: as $x\ge0$, I will replace $x$ by $z^2$ to get rid of the half-exponents. As the relation $x=z^2$ is monotonous, this will not cause trouble.

Now,

$$\left(\frac{z}{(z^3+2)^2}\right)'=\frac{(z^3+2)^2-6z^3(z^3+2)}{(z^3+2)^4}$$ and after simplification the numerator is

$$2-5z^3.$$

Hence

$$z\ge\sqrt[3]{\frac 25}$$ or $$x\ge\left(\frac 25\right)^{2/3}.$$

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  • $\begingroup$ Thx that really helped. Those half exponents really got in the way when tryng the find the derivative so this little trick really helped me so thnx $\endgroup$ – Jasper De Zoete Apr 4 at 22:36

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