increasing/decreasing function help?

ok so I have a supposedly simple function but I can't seem to understand it;

y=x^(1.5)
dy/dx=1.5x^0.5

show y is an increasing function?

..so all values of x are meant to give a positive y values meaning it's increasing but I don't think it does? and how do I show this?

thanks

What level is this? A level, university? The answer will depend on what you already know.

Anyway, you need to show either that, for +ve x:

a) $dy/dx >0 \Rightarrow x^{0.5}=\sqrt{x} > 0$, or

b) $a > b \Rightarrow a^{3/2} > b^{3/2}$

But the first is essentially trivial since the +ve square root is +ve by definition, and the second can be done by, say:

1. using the fact that $\log$ is an increasing function (what is its derivative?).

2. considering, for $a > b >0$, $a^{3/2}-b^{3/2}=\sqrt{a^3}-\sqrt{b^3}$, rationalising the numerator using the difference-of-two-squares, and proceeding from there to show that the expression is +ve.