Let \(\mathrm{f}\) be a continuous function defined for \(0 \leqslant x \leqslant 1\). Show that
\[\int_0^1 \mathrm{f}(\sqrt{x})\,\mathrm{d}x = 2\int_0^1 x\,\mathrm{f}(x)\,\mathrm{d}x\,.\]
Let \(\mathrm{g}\) be a continuous function defined for \(0 \leqslant x \leqslant 1\) such that
\[\int_0^1 \big(\mathrm{g}(x)\big)^2\,\mathrm{d}x = \int_0^1 \mathrm{g}(\sqrt{x})\,\mathrm{d}x - \frac{1}{3}\,.\]
Show that \(\displaystyle\int_0^1 \big(\mathrm{g}(x) - x\big)^2\,\mathrm{d}x = 0\) and explain why \(\mathrm{g}(x) = x\) for \(0 \leqslant x \leqslant 1\).
Let \(\mathrm{h}\) be a continuous function defined for \(0 \leqslant x \leqslant 1\) with derivative \(\mathrm{h}'\) such that
\[\int_0^1 \big(\mathrm{h}'(x)\big)^2\,\mathrm{d}x = 2\mathrm{h}(1) - 2\int_0^1 \mathrm{h}(x)\,\mathrm{d}x - \frac{1}{3}\,.\]
Given that \(\mathrm{h}(0) = 0\), find \(\mathrm{h}\).
Let \(\mathrm{k}\) be a continuous function defined for \(0 \leqslant x \leqslant 1\) and \(a\) be a real number, such that
\[\int_0^1 \mathrm{e}^{ax}\big(\mathrm{k}(x)\big)^2\,\mathrm{d}x = 2\int_0^1 \mathrm{k}(x)\,\mathrm{d}x + \frac{\mathrm{e}^{-a}}{a} - \frac{1}{a^2} - \frac{1}{4}\,.\]
Show that \(a\) must be equal to \(2\) and find \(\mathrm{k}\).