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]

#Calculus

Easy Math Editor

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Comments

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 - 5 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$