
Easy
Find all the critical points of the following functions. Say whether the critical points are local maxima, local minima,
saddle points, or otherwise.
 \(f(x,y) = x^42x^2+y^36y\)
 \(f(x,y) = (x1)(x^2y^2)\)
 \(f(x,y) = x^2y^2(1xy)\)
 \(f(x,y) =(2x^2+y^2)e^{x^2y^2}\)
 \(f(x,y,z) = xyz(4xyz)\)

Easy
Find the extreme values of \(f(x,y,z) = x^2 + 2y^2 + 3z^2\) on the unit
sphere, \(x^2 + y^2 + z^2 = 1\).

Easy
What conditions on \(a\), \(b\), and \(c\) guarantee that \(f(x,y) = ax^2
+ bxy + cy^2\) has local max, a local min, or a saddle point at \((0,0)\)?

Easy
What is the volume of the largest box which can be fit inside of the
sphere \(x^2 + y^2 + z^2 = 1\)?

Medium
Use the method of Lagrange multipliers to find the smallest distance
between the parabola \(y = x^2\) and the line \(y = x1\).

Medium
Let \(f(x,y) = (yx^2)(y2x^2)\). Show that the origin is a degenerate
critical point of \(f\). Prove that \(f\) restricted to any line through the
origin has a local minimum, but \(f\) does not have a local minimum at the
origin. (Hint: Consider the regions where \(f >0\) and \(f< 0\)).

Medium
Find the minimum of the function \(f(x,y,z) = x^2 + y^2 + z^2\) on the
surface \(x^2+y^2z^2 = c\), where \(c \in \mathbb{R}\). Hint: You'll need to
break this into cases, depending on the value of \(c\).

Hard
Let \(A:\mathbb{R}^{n}\to\mathbb{R}^{n}\) be a symmetric linear map and define \(f:\mathbb{R}^n\to\mathbb{R}\) by \(f(\mathbf{x})=A\mathbf{x}\cdot\mathbf{x}\). Show that
on the set \(S = \{v \in \mathbb{R}^{n}: \vert\vert v\vert\vert = 1\}\), the
maximum and minimum of \(f\) are the largest and smallest eigenvalues of \(A\),
respectively.