Question

Give an upper bound and a lower bound for the expression 1/(a^(4)+3a^(2)+1) if a∈R

Answer 1

The function $f(a)=\dfrac{1}{a^4+3a^2+1}$ satisfies $0<f(a)\leq1$ for all $a\in\mathbb{R}$. Moreover, $f(0)=1$ and $f(a)$ tends to 0 as $a$ tends to $\pm\infty$. Then the upper bound of $\dfrac{1}{a^4+3a^2+1}$ is 1 and its lower bound is 0.