```{r data generation, echo = FALSE, results = "hide"}
coef <- sample(c(2:9, -(2:9)), 3, replace = TRUE)
x <- sample(c(-5:5), 2, replace = TRUE)
H <- matrix(c(2 * coef[1], coef[2], coef[2], 2 * coef[3]),
  nrow = 2, ncol = 2)

ix <- sample(1:4, 1, prob=c(0.35, 0.15, 0.15, 0.35))
ixt <- c("upper left", "upper right", "lower left", "lower right")[ix]
ixn <- c("11", "12", "21", "22")[ix]
sol <- H[ix]

err <- unique(H[-ix])
err <- err[err != sol]
sc <- num_to_schoice(sol, wrong = err, range = -25:25, method = "delta", delta = 1, digits = 0)

plus <- ifelse(coef < 0, "", "+")
```

Question
========
Compute the Hessian of the function
$$
\begin{aligned}
  f(x_1, x_2) = `r coef[1]` x_1^{2} `r plus[2]` `r coef[2]`  x_1  x_2 `r plus[3]` `r coef[3]`  x_2^{2}
\end{aligned}
$$
at $(x_1, x_2) = (`r x[1]`, `r x[2]`)$.
What is the value of the `r ixt` element?

```{r questionlist, echo = FALSE, results = "asis"}
answerlist(sc$questions, markup = "markdown")
```

Solution
========
The first-order partial derivatives are 
$$
\begin{aligned}
  f'_1(x_1, x_2) &= `r H[1,1]` x_1 `r plus[2]` `r H[1,2]` x_2  \\
  f'_2(x_1, x_2) &= `r H[2,1]` x_1 `r plus[3]` `r H[2,2]` x_2
\end{aligned}
$$
and the second-order partial derivatives are
$$
\begin{aligned}
  f''_{11}(x_1, x_2) &= `r H[1,1]`\\
  f''_{12}(x_1, x_2) &= `r H[1,2]`\\
  f''_{21}(x_1, x_2) &= `r H[2,1]`\\
  f''_{22}(x_1, x_2) &= `r H[2,2]`
\end{aligned}
$$

Therefore the Hessian is
$$
\begin{aligned}
  f''(x_1, x_2) = `r toLatex(H, escape = FALSE)`
\end{aligned}
$$
independent of $x_1$ and $x_2$. Thus, the `r ixt` element is:
$f''_{`r ixn`}(`r x[1]`, `r x[2]`) = `r sol`$.


```{r solutionlist, echo = FALSE, results = "asis"}
answerlist(ifelse(sc$solutions, "True", "False"), markup = "markdown")
```

Meta-information
================
extype: schoice
exsolution: `r mchoice2string(sc$solutions)`
exname: Hessian