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lato1 · Dec 5, 2011, 11:36 PM

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.

Aurora2000 · Dec 6, 2011, 02:24 AM

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.

lato1 · Dec 6, 2011, 05:23 AM

Thank you so much u explained clearly