How does one answer these type of question:
(a) If f is continuous, decreasing function on [1, $\infty$) and $\lim\limits_{x\to\infty} f(x) = 0$, then $\int_{1}^{\infty} f(x) dx$ is convergent.
(b) If $f(x) \leq g(x)$ for all x and $\int_{0}^{\infty} f(x)dx$ diverges, then $\int_{0}^{\infty} g(x)dx$ also diverges.
(c) If $\int_{7}^{\infty}f(x)dx$ convereges and let $c \in R$, then $\int_{7}^{\infty} cf(x)dx$ also converges.
For c) i just argued that the constant doesn't matter because of the integral property of putting the constant out. So if it diverges or converges you just multiply the constant. So true. Bad explanation i think
(d) Let $a \in R$. If $0 \leq f(x) \leq g(x)$ for all $x \in [a,\infty)$ and $\int_{a}^{\infty} g(x)dx$ diverges, then $\int_{a}^{\infty} f(x)dx$ also diverges.
i have no idea for d.
I was reading theorems in this section in my textbook but none of them really helped me figure this out.
(a) is false. Take for example $f:x\mapsto\frac 1x$
(b) is also false (and would be true with the additional assumption that $f(x)\ge0$ for all $x\ge0$). Consider for example $f:x\mapsto-1$ and $g:x\mapsto\exp(-x)$.