Additional exercises 2

Exercise

Consider the scalar field

$$f(x,y)=y^4+3x^2-4xy-5y+81.$$

Calculate the Hessian matrix at the point $(2,1)$.

Exercise

Consider the scalar field $f(x,y)=8\sin x\sin y$. Calculate the second order Taylor approximation of $f$ at the point $(\frac{\pi }{2},\frac{\pi }{2})$. Use that:

$$f(\mathbf{x})\approx f(\mathbf{a})+\mathrm{\nabla }f(\mathbf{a})\cdot (\mathbf{x}-\mathbf{a})+\frac{1}{2}(\mathbf{x}-\mathbf{a})\mathbf{H}f(\mathbf{a})(\mathbf{x}-\mathbf{a}{)}^{\mathrm{T}}.$$

Exercise

Locate and classify the extrema points of the scalar field

$$f(x,y)=x^4+3xy+2y^2+1.$$

Exercise

Locate and classify the stationary points of the scalar field

$$f(x,y)=2x^2-xy-3y^2-3x+7y+11.$$

Exercise

Locate and classify the stationary points of the scalar field

$$f(x,y,z)=2e^{(x-1{)}^2}(y^4-4yz+2z^2).$$

Note: this problem is rather harder than the previous ones and harder than anticipated in exams.

Exercise

Find the absolute minimum and absolute maximum of $f(x,y)=(9x^2-1)(1+4y)$ on the rectangle given by $-2\le x\le 3$, $-1\le y\le 4$.

Exercise

Apply the Lagrange multiplier method to find the points on the curve $x^2+xy+y^2=21$ which are closest / furthest to the origin. Out of curiosity, sketch the curve and identify the extrema points found.

Exercise

Apply the Lagrange multiplier method to find the points on the curve of intersection of the two surfaces

$$x^2-xy+y^2-z^2=4,x^2+y^2=4$$

which are nearest the origin.

Exercise

Apply the Lagrange multiplier method to find the extrema of $f(x,y,z)=yz+xy$ subject to the constraints $xy=1$ and $y^2+z^2=1$.