# Let f(x)=(x^2+3)/(2x) for x not equal to 0. Show that if x>3^(1/2), then f(x)>3^(1/2)

• July 3rd 2012, 02:02 PM
juanma101285
Let f(x)=(x^2+3)/(2x) for x not equal to 0. Show that if x>3^(1/2), then f(x)>3^(1/2)
Hi, this exercise is giving headaches :(. I only need help with Part A because I know part B and have an idea of how to solve other exercises similar to part C. Thanks a lot!

The whole exercise says:

"Let $f(x)=(x^2+3)/(2x)$ for $x \neq 0$. Define a sequence of real numbers $x_n$ by

$x_{n+1}=f(x_n)$ for $n \ge 1$, $x_1=2$.

A) Show that if $x>\sqrt{3}$, then $f(x)>\sqrt{3}$.
B) Show that if $x>\sqrt{3}$, then $x>f(x)$.
C) Conclude that the sequence $x_n$ converges."
• July 3rd 2012, 02:09 PM
richard1234
Re: Let f(x)=(x^2+3)/(2x) for x not equal to 0. Show that if x>3^(1/2), then f(x)>3^(
A) Note that $f(\sqrt{3}) = \sqrt{3}$. Furthermore, $f'(x) = \frac{2x(2x) - 2(x^2 + 3)}{4x^2} = \frac{2x^2 - 6}{4x^2}$, which is always positive when $x > \sqrt{3}$. Hence, the minimal value of $f(x)$ is $f(\sqrt{3})$, or $\sqrt{3}$.

Therefore $f(x) > \sqrt{3}$ for all $x > \sqrt{3}$.
• July 3rd 2012, 02:37 PM
juanma101285
Re: Let f(x)=(x^2+3)/(2x) for x not equal to 0. Show that if x>3^(1/2), then f(x)>3^(
Thanks! I didn't even think it could be solved so easily *sigh*