A curve has the equation
y=ax2y = a^{x^2}y=ax2
where a a\,a is a constant greater than 1.
Show that dydx=2xax2lna\dfrac{dy}{dx} = 2xa^{x^2}\ln adxdy=2xax2lna.
The tangent to the curve at the point (1,a)(1, a)(1,a) passes through the point (12,0)\displaystyle \left(\frac{1}{2}, 0\right)(21,0). Find the value of aaa, giving your answer in exact form.
By considering d2ydx2\dfrac{d^2y}{dx^2}dx2d2y, show that the curve is convex for all values of xxx.