Problem Set 10

  1. Easy
    Find all the critical points of the following functions. Say whether the critical points are local maxima, local minima, saddle points, or otherwise.
    1. \(f(x,y) = x^4-2x^2+y^3-6y\)
    2. \(f(x,y) = (x-1)(x^2-y^2)\)
    3. \(f(x,y) = x^2y^2(1-x-y)\)
    4. \(f(x,y) =(2x^2+y^2)e^{-x^2-y^2}\)
    5. \(f(x,y,z) = xyz(4-x-y-z)\)

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

  3. 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 \((0,0)\)?

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

  5. Medium
    Use the method of Lagrange multipliers to find the smallest distance between the parabola \(y = x^2\) and the line \(y = x-1\).

  6. Medium
    Let \(f(x,y) = (y-x^2)(y-2x^2)\). 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 \(f< 0\)).

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

  8. 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.

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