Checking My Work: Vanishing Value for \partial F/\partial x | Easy Question

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For what value does \frac{\partial F}{\partial x} vanish?

Working is in the attached pictures, easy question just want to make sure I haven't stuffed up

Thank you =D

Mod Edit: Fixed LaTeX. Tip: leave a space between delta and whatever comes after it. If you don't, the LaTeX rendering system doesn't recognize deltaF and similar. Also, for partial derivatives, use \ partial F and \ partial x, not \ delta F and \ delta x.
 

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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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