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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]

Note by Akhilesh Prasad
1 year, 10 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 \( x_o \)

Siddhartha Srivastava - 1 year, 10 months ago

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