Show that every line $y=mx$ intersects the curve $y^2+\int _{ 0 }^{ x }{ f(t)dt=2!}$

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  • If $\int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then every line $y=mx$ Intersect $ y^2+\int_{0}^{x}f(t)dt=2$

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Solutions Collecting From Web of "Show that every line $y=mx$ intersects the curve $y^2+\int _{ 0 }^{ x }{ f(t)dt=2!}$"

Set : $\displaystyle g(x)= (mx)^2+\int_0^x f(t)\ \mathrm{d}t$, $g$ is continuous since it is the sum of two continuous

(and also differentiable) functions.

We have : $g(0)=0$, and : $\displaystyle \lim_{x\to \infty} g(x)= +\infty$, for any $m\in \mathbb{R}$ .

So the equation : $g(x)=\alpha$ has a solution for any $\alpha \geq 0$.

Here we have $\alpha =2$

Consider a function
$$F(x)=y^2+\int _{ 0 }^{ x}f(t)dt-2$$.

$F(x)$ is continuous as $f(x)$ is continuous.

Putting $y=mx$ in the equation of F(x).

$$F(x)={m^2x^2}+\int _{ 0 }^{ x}f(t)dt-2$$

Clearly $F(0)=-2$

$F(x)\rightarrow\infty$ as $x\rightarrow\infty$ so somewhere F(x) must exceed 0 and hence somewhere F(x) is indeed 0.

So y=mx surely intersects the line!

P.S:My solution is quite similar to @juantheron though :-)!