Homework 12: Extrema

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Graded Problems

Consider the function $f(x,y) = (y^2-x)(2y^2-x)$ .

(a): Show that $f(x,y)$ has a critical point at the origin.
(b): Show that $f(x,y)$ has a local minimum along any line through the origin. That is, show that for constant $(a,b)\neq (0,0)$ , that the function $g(t)=f(at,bt)$ has a local minimum at $t=0$ .
(c): Show that $f(x,y)$ does not have a local minimum at the origin.

(a): Let $x_1,x_2,...,x_n$ be nonnegative numbers such that their sum is constant. That is, $x_1+x_2+\cdots +x_n=C$ for some constant $C$ . Show that the product $x_1x_2\cdots x_n$ is a maximum if and only if $x_1=x_2=\cdots =x_n = C/n$ .
(b): Using the result of part (a), show that if $x_1,x_2,...,x_n$ are nonnegative numbers such that $x_1+x_2+\cdots +x_n=1$ , then $x_1x_2\cdots x_n\leq 1/n^n$ .

Professional Problem

Final draft of your project, worth two professional problems.

Completion Packet

Find and classify all critical points of each function. When using the Hessian fails to classify points, find another method.

(a): $f(x,y) = x^3 + y^3 + 12xy$
(b): $f(x,y) = x^2+y^2+\frac {1}{x^2y^2}$ , for $xy\neq 0$
(c): $f(x,y) = (x+y)^2+x^4$
(d): $f(x,y) = (x+y)e^{-xy}$
(e): $f(x,y,z) = 2x^2+y^2+z^2-xz+xy$
(f): $f(x,y,z) = xy+xz$

What are the conditions on $a,b,c$ for $f(x,y) = ax^2+bxy+cy^2$ to have a...

(a): ...local minimum at the origin?
(b): ...local maximum at the origin?
(c): ...saddle point at the origin?

For nonzero constants $a$ and $b$ , consider the function $f(x,y) = ax^{-1}+by^{-1}+xy$ .

(a): Find the (single) critical point of this function.
(b): What are the conditions on $a$ and $b$ for the critical point to be a local minimum? A local maximum? A saddle point?

Find the shortest distance between a point on the surface $(x-2)^2+(y-3)^2+z^2=1$ and the origin in $\mathbb {R}^3$ .

(a): Let $x_1,x_2,...,x_n$ be positive numbers such that their product is constant. That is, $x_1x_2\cdots x_n=C$ for some constant $C$ . Show that the sum $x_1+x_2+\cdots +x_n$ is a local minimum if and only if $x_1=x_2=\cdots =x_n$ .
(b): The local minimum which you found in part (a) is, in fact, an absolute minimum (you do not need to show this). Using this fact, show that if $x_1,x_2,...,x_n$ are positive numbers such that $x_1x_2\cdots x_n=1$ , then $x_1+x_2+\cdots +x_n\geq n$ .