数学代写|MATH3404 Optimisation Theory

3.3 Find the maximum value of $f(x, y)=x^2 y$, if $x$ and $y$ are restricted to positive real numbers for which $6 x+5 y=45$.
Write
$$
f(x, y)=x x y=\frac{1}{45}(3 x)(3 x)(5 y),
$$
and
$$
45=6 x+5 y=3 x+3 x+5 y .
$$
3.4 Find the smallest value of
$$
f(x)=5 x+\frac{16}{x}+21,
$$
over positive $x$.
Just focus on the first two terms.

3.5 Find the smallest value of the function
$$
f(x)=\sqrt{x^2+y^2},
$$
among those values of $x$ and $y$ satisfying $3 x-y=20$.
By Cauchy’s Inequality, we know that
$$
20=3 x-y=(x, y) \cdot(3,-1) \leq \sqrt{x^2+y^2} \sqrt{3^2+(-1)^2}=\sqrt{10} \sqrt{x^2+y^2},
$$
with equality if and only if $(x, y)=c(3,-1)$ for some number $c$. Then solve for $c$.
3.6 Find the maximum and minimum values of the function
$$
f(x)=\sqrt{100+x^2}-x
$$
over non-negative $x$.

The derivative of $f(x)$ is negative, so $f(x)$ is decreasing and attains its maximum of 10 at $x=0$. As $x \rightarrow \infty, f(x)$ goes to zero, but never reaches zero.

3.7 Multiply out the product
$$
(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)
$$
and deduce that the least value of this product, over non-negative $x, y$, and $z$, is 9 . Use this to find the least value of the function
$$
f(x)=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}
$$
over non-negative $x, y$, and $z$ having a constant sum $c$.
See the more general Exercise 3.9 .
3.8 The harmonic mean of positive numbers $a_1, \ldots, a_N$ is
$$
H=\left[\left(\frac{1}{a_1}+\ldots+\frac{1}{a_N}\right) / N\right]^{-1} .
$$
Prove that the geometric mean $G$ is not less than $H$.
Use the AGM Inequality to show that $H^{-1} \geq G^{-1}$.

3.9 Prove that
$$
\left(\frac{1}{a_1}+\ldots+\frac{1}{a_N}\right)\left(a_1+\ldots+a_N\right) \geq N^2
$$
with equality if and only if $a_1=\ldots=a_N$.
When we multiply everything out, we find that there are $N$ ones, and $\frac{N(N-1)}{2}$ pairs of the form $\frac{a_m}{a_n}+\frac{a_n}{a_m}$, for $m \neq n$. Each of these pairs has the form $x+\frac{1}{x}$, which is always greater than or equal to two. Therefore, the entire product is greater than or equal to $N+N(N-1)=N^2$, with equality if and only if all the $a_n$ are the same.
3.10 Show that the Equation $S=U L U^T$, can be written as
$$
S=\lambda_1 u^1\left(u^1\right)^T+\lambda_2 u^2\left(u^2\right)^T+\ldots+\lambda_N u^N\left(u^N\right)^T
$$
and
$$
S^{-1}=\frac{1}{\lambda_1} u^1\left(u^1\right)^T+\frac{1}{\lambda_2} u^2\left(u^2\right)^T+\ldots+\frac{1}{\lambda_N} u^N\left(u^N\right)^T
$$

This is just algebra.