The curve \(C\) is defined parametrically by
\(x=4 t-t^{2} \quad \text { and } \quad y=1-\mathrm{e}^{-t}\)
where \(0 \leqslant t\lt 2\). Show that at all points of \(C\),
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{(t-1) \mathrm{e}^{-t}}{4(2-t)^{3}}\)

Show that the mean value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac{7}{4}\) is
\(\frac{4 e^{-\frac{1}{2}}-3}{21} .\)