Question

If $f\left(x\right)$ is continuous function such that ${\int }_0^{x}f\left(t\right)dt\to \infty$ as $x\to \infty$, show that every line $y=mx$ intersect the curve $y^2+{\int }_0^{x}f\left(t\right)dt=a$ where $a\in R^{+}$

Answer 1

We have to shoe that there exits same $x$ such that $m^2{x}^2+{\int }_0^{x}f\left(t\right)dt=a(a\in R^{+})$ ...(1)
Consider the function
$g\left(x\right)=m^2{x}^2+{\int }_0^{x}f\left(t\right)dt$
Since $f$ is a continuous function, therefore $g$ is also a continous function. Also $g\left(0\right)=0$ and $g\left(x\right)\to \infty$ as $x$ be same $x\in (0,\infty )$, such that $g\left(x\right)=a(a\in R^{+})$
Hence, for every real $m$, there exists same $a(a\in R^{+})$ that satisfies equation (1)