<s>[INST] How to determine the sign of : I've got some trouble: I don't know how to determine the sign of : 
$$\frac{4t^2- \sqrt t}{t}$$
for $t > 0$.
Thank you in advance
 [/INST] Since $\sqrt{t}$ is only defined when $t \ge 0$, we can only consider inputs from that range. 
Now we have to consider the sign of both the numerator and the denominator. The fraction will be positive whenever the sign of $4t^2 - \sqrt{t}$ matches that of $t$. Clearly though, $t$ is only positive when $t \ge 0$, so we see that the sign of the numerator determines the sign of the fraction.
Consider the equation $4t^2 - \sqrt{t} = 0.$ We can solve this as follows:
$$4t^2 = \sqrt{t}$$
$$(4t^2)^2 = t$$
$$16t^4 = t$$
$$16t^4 - t = 0$$
$$t(16t^3 - 1) = 0.$$
So the numerator equals zero whenever $t = 0$ (which is not on our interval) or $16t^3 - 1 = 0$. Solving this last equation we see that the numerator equals zero when $t = (\frac{1}{16})^\frac{1}{3}$.
We now pick a point larger than this value, say t = 1. Plugging in, we see that $4(1)^2 - \sqrt{1} = 4 - 1 = 3 > 0.$ Thus, whenever $t > (\frac{1}{16})^\frac{1}{3}$, the numerator, and the fraction, are positive. Picking a point between $0$ and $(\frac{1}{16})^\frac{1}{3}$ will reveal that the numerator is negative on the interval $(0, (\frac{1}{16})^\frac{1}{3}).$ For instance, $t = \sqrt{\frac{1}{16}}.$ Plugging in this value to the numerator gives
$$4(\sqrt{\frac{1}{16}})^2 - \sqrt{\sqrt{\frac{1}{16}}} = $$
$$4(\frac{1}{16}) - (\frac{1}{16})^{\frac{1}{4}} = $$
$$ \frac{4}{16} - \frac{1}{2} < 0.$$
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