Partial Derivatives & the Gradient · Superchallenge problem

For real $a$ and $b$ let $V(a,b) = \displaystyle\max_{t\in\mathbb{R}}\left(at + bt^2 - \frac{t^4}{4}\right)$. Near $(a,b) = \left(6, \dfrac12\right)$ this maximum is attained at a single point $t^*(a,b)$ that depends smoothly on $a$ and $b$. Find $\dfrac{\partial^2 V}{\partial a\,\partial b}$ at $\left(6, \dfrac12\right)$.