12 A curve has equation \(y = a ^ { 3 x ^ { 2 } }\), where \(a\) is a constant greater than 1 .
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x a ^ { 3 x ^ { 2 } } \ln a\).
- The tangent at the point \(\left( 1 , a ^ { 3 } \right)\) passes through the point \(\left( \frac { 1 } { 2 } , 0 \right)\).
Find the value of \(a\), giving your answer in an exact form.
- By considering \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) show that the curve is convex for all values of \(x\).
\section*{OCR}
\section*{Oxford Cambridge and RSA}