{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(scquestions, 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 ofx_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