# 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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