suppose that $f,g : \mathbb{R}$ to $\mathbb{R}$ are differentiable functions such that f is strictly increasing and g is strictly decreasing. Define $p(x) = f(g(x)$ and $q(x) = g(f(x)),\forall x\epsilon\Bbb{R}$. then for t>0, the sign of $$\int_{o}^{t}p'(x)(q'(x)-3)dx $$ is
a. positive b. negative c. dependent on t d. dependent on f and g
The answer is a): it is easy to verify that $p$ and $q$ are both decreasing. Hence $p'$ and $q'$ are both negative. Hence the integral is positive.