# Trying this problem for sometime

If $$f$$ is a continuous function with $$\displaystyle \int_0^x f(t) \, dt \to \infty$$ as $$|x| \to \infty$$, then show that every line $$y = mx$$ intersects the curve $$\displaystyle y^2 + \int_0^x f(t) \, dt = 2$$.

###### Source: [I. I. T. 91]

2 years, 6 months ago

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Probably wrong. But here's my attempt.

Let $$F(x) = \displaystyle \int_0^x f(t) dt$$

Then, when $$x = 0$$, $$F(x) = 0$$, and when $$|x| \rightarrow \infty$$, $$F(x) \rightarrow \infty$$.

Now, let $$G(x) = m^2x^2 + F(X)$$. Now, when, when $$x = 0$$, $$G(x) = 0$$, and when $$|x| \rightarrow \infty,$$, $$G(x) \rightarrow \infty$$.

By the IVT, $$G(x_o) = 2$$ for some $$x_o$$

- 2 years, 6 months ago