If" f, is a continuous function with `int_0^x f(t) dt->oo` as `|x|->oo`then show that every line `y = mx` intersects the curve `y^2 + int_0^x f(t) dt = 2`

Explanation

We are given that `f` is a continuous function and
`int_(0)^(x)f(t)dt to oo` as `|x|to oo`
We have to show that every line `y=mx` intgersects the curve
`y^(2)+int_(0)^(x)f(t)dt=2`
If possible, let `y=mx` intersects the given curve. Then substituting `y=mx` in the curves, we get
`m^(2)x^(2)+int_(0)^(x)f(t)dt=2`
Consider `F(x)=m^(2)x^(2)+int_(0)^(x)f(t)dt-2`
Then `F(x)` is a continuous function as `f(x)` is given to be continuous.
Also `F(x)to oo` as `|x|to oo`
But `F(0)=-2`
So the graph of `y=F(x)` must corss `x` -axis to reach infinity as `|x|` approaches infinity.
Hence `y=mx` intersects the given curves.