Proving Inequalities with Positive Real Numbers

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1. c, d positive real numbers and c+d=1 prove that (c+1/c)2+(d+1/d)2>=25/2
2. Prove the previous also formulate and prove a generalization for n
3. I first tried to expand but got stuck, I thought of using Binomial but also got stuck. I tried to do (a+b)2 were a=(c+1/c) and b=(d+1/d) but also nothing
Any help is appreciated
 
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Well, if c is greater than d, and c=1-d, then 1-d > d. Add d to both sides and divide by 2 to get that .5>d. So you know that .5>d, and similar processes lead you to the fact that c>.5 . I have a feeling that this might help you, but I'm not sure.
 

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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