Let $a,$ $b,$ $c,$ and $d$ be positive real numbers such that $a + b + c + d = 10.$  Find the maximum value of $ab^2 c^3 d^4.$
Let's use AM-GM inequality. AM-GM inequality is $\frac{a_1 + a_2 + \cdots + a_n}{n} \ge \sqrt[n]{a_1a_2\cdots a_n}$. That's right. Let's apply it to this problem. AM-GM inequality on the four variables $a$, $b$, $c$, and $d$ gives us $\frac{a+b+c+d}{4} \ge \sqrt[4]{abcd}$. And we know that $a+b+c+d=10$. Then we have $\sqrt[4]{abcd} \le \frac{10}{4}$, so $abcd \le \frac{10^4}{4^4}$. Umm... that's not quite what we want. We want to maximize $ab^2c^3d^4$. Right. We can manipulate the AM-GM inequality to suit our needs. We just have to make sure that the product on the right-hand side contains $ab^2c^3d^4$. In other words, we want $a_1$, $a_2$, $\cdots$, $a_n$ to be $a$, $b$, $c$, $d$, $b$, $c$, $c$, $d$, $d$, $d$. Then AM-GM inequality tells us that $\frac{a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8+a_9+a_{10}}{10} \ge \sqrt[10]{a_1a_2a_3a_4a_5a_6a_7a_8a_9a_{10}}$. We want to maximize $f(a,b,c,d)=ab^2c^3d^4$ subject to the constraint $g(a,b,c,d)=a+b+c+d-10=0$. We need to compute $\nabla f$ and $\nabla g$. $\nabla f=\langle b^2c^3d^4, 2ab^1c^3d^4, 3ab^2c^2d^4, 4ab^2c^3d^3 \rangle$ And $\nabla g=\langle 1, 1, 1, 1 \rangle$. Now we need to solve the equation $\nabla f = \lambda \nabla g$. So we get the system of equations $$b^2c^3d^4=\lambda \\ 2ab^1c^3d^4=\lambda \\ 3ab^2c^2d^4=\lambda \\ 4ab^2c^3d^3=\lambda \\ a+b+c+d=10$$. We can see that $\lambda=0$ is not a valid solution. So we divide both sides of the first equation by $b^2c^3d^4$ and both sides of the second equation by $2ab^1c^3d^4$ and both sides of the third equation by $3ab^2c^2d^4$ and both sides of the fourth equation by $4ab^2c^3d^3$ to get the system of equations. That's right. The first four equations tell us that $a:b:c:d=1:2:3:4$. That's right. And the last equation tells us that $1+2+3+4=10$. So $a=1$, $b=2$, $c=3$, and $d=4$ is indeed the solution. That's right.

