Vector Calculus | Part IA, 2002

Two independent variables x1x_{1} and x2x_{2} are related to a third variable tt by

x1=a+αt,x2=b+βt,x_{1}=a+\alpha t, \quad x_{2}=b+\beta t,

where a,b,αa, b, \alpha and β\beta are constants. Let ff be a smooth function of x1x_{1} and x2x_{2}, and let F(t)=f(x1,x2)F(t)=f\left(x_{1}, x_{2}\right). Show, by using the Taylor series for F(t)F(t) about t=0t=0, that

f(x1,x2)=f(a,b)+(x1a)fx1+(x2b)fx2+12((x1a)22fx12+2(x1a)(x2b)2fx1x2+(x2b)22fx22)+\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 x1=a,x2=bx_{1}=a, x_{2}=b.

Hence show that a stationary point (a,b)(a, b) of f(x1,x2)f\left(x_{1}, x_{2}\right) is a local minimum if

H11>0,detHij>0H_{11}>0, \quad \operatorname{det} H_{i j}>0

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

Find two local minima of

f(x1,x2)=x14x12+2x1x2+x22f\left(x_{1}, x_{2}\right)=x_{1}^{4}-x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}

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