Proofs & Rational Number Btwn $\sqrt2$ & $\sqrt2 + \frac{1}{1000}$

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1. Show that between any two real numbers there is a rational number.
You may assume that for each a>0 and b>0 in R there is a positive integer satisfying b<ma.

2.Find the intervals on which f(x) = (3x -2)(x 1)(x + 1) is positive, and
the intervals on which it is negative.

3.Find a rational number p/q between root of 2 and root of 2 + (1/1000)
 
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Hi KelvinMa! Welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 
thx tiny-tim : D
i really don't have any ideas about the first question and the third question
i ve looked up on my notes and i couldn't find any clues
the second question i ll find the max and min points but is there any other way to do it?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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