Two independent variables $x_{1}$ and $x_{2}$ are related to a third variable $t$ by

$$x_{1}=a+\alpha t, \quad x_{2}=b+\beta t,$$

where $a, b, \alpha$ and $\beta$ are constants. Let $f$ be a smooth function of $x_{1}$ and $x_{2}$, and let $F(t)=f\left(x_{1}, x_{2}\right)$. Show, by using the Taylor series for $F(t)$ about $t=0$, that

$$\begin{gathered} f\left(x_{1}, x_{2}\right)=f(a, b)+\left(x_{1}-a\right) \frac{\partial f}{\partial x_{1}}+\left(x_{2}-b\right) \frac{\partial f}{\partial x_{2}} \\ +\frac{1}{2}\left(\left(x_{1}-a\right)^{2} \frac{\partial^{2} f}{\partial x_{1}^{2}}+2\left(x_{1}-a\right)\left(x_{2}-b\right) \frac{\partial^{2} f}{\partial x_{1} \partial x_{2}}+\left(x_{2}-b\right)^{2} \frac{\partial^{2} f}{\partial x_{2}^{2}}\right)+\ldots \end{gathered}$$

where all derivatives are evaluated at $x_{1}=a, x_{2}=b$.

Hence show that a stationary point $(a, b)$ of $f\left(x_{1}, x_{2}\right)$ is a local minimum if

$$H_{11}>0, \quad \operatorname{det} H_{i j}>0$$

where $H_{i j}=\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}$ is the Hessian matrix evaluated at $(a, b)$.

Find two local minima of

$$f\left(x_{1}, x_{2}\right)=x_{1}^{4}-x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}$$