"Find the range" → solve the quadratic inequality (sketch the parabola). "Show that" → argue the sign directly.

Reflex: Differentiate, then ask: where is this positive/negative?

For $x>0$, each term is positive, so $\frac{dy}{dx}>0$ — increasing, shown. Reflex: Name each piece's sign, then conclude on $\frac{dy}{dx}$.

$y$ increasing → sign of $\frac{dy}{dx}$. Gradient increasing → sign of $\frac{d^2y}{dx^2}$. Here $f^{(2)}(x)<0$ → the gradient itself is decreasing.

Reflex: The quantity you differentiate is the one whose monotonicity you're judging.