let f and g be differentiable functions on (-infinity;infinity) such that f′(a) > 0 for all a and g′(b) < 0
for all b > 0. Let h(x) = f(g(x^2)). Find the open intervals on which h is increasing or decreasing.

The conditions on f and g do not display correctly. I will assume there are
f'(a)>0\ \forall a,\ g'(b)<0\ \forall b>0
Anyway, as they are differential functions, to find where f(g(x^2)) is increasing or decreasing you can compute the first derivative in x :

\frac{d}{dx}f(g(x^2))=f'(g(x^2))g'(x^2)2x

You have g(x^2), x^2\geq 0 thus g'(x^2)\leq 0; moreover, f'(a)>0 \ \forall a implies f'(g(x^2))>0 . So this means that \frac{d}{dx}f(g(x^2)) is not negative for x negative, not positive for x positive.

Thank you so much u explained clearly