Question
========
Consider the following mathematical definitions for a linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ represented by matrix $A$, and a continuous function $f(x)$:

$A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$
<br><br>
$f(x) = \int_{0}^{x} (3t^2 - 12) dt$
<br><br>

Which of the following mathematical statements are **TRUE**?
*(Select all that apply)*

```{r questionlist, echo = FALSE, results = "asis"}
exams::answerlist(c(
  "The determinant of matrix $A$ is equal to $10$.",
  "The eigenvalues of matrix $A$ are $\\lambda_1 = 5$ and $\\lambda_2 = 2$.",
  "Matrix $A$ is a singular matrix (non-invertible).",
  "The first derivative of the function is $f'(x) = 3x^2 - 12$.",
  "The function $f(x)$ has a local maximum at $x = 2$."
), markup = "markdown")
```

Solution
========
1. **True:** Det($A$) = $(4 \\times 3) - (2 \\times 1) = 12 - 2 = 10$.
2. **True:** The trace is $7$ and determinant is $10$, which correspond to eigenvalues $5$ and $2$.
3. **False:** Because the determinant is $10$ (not $0$), the matrix is invertible.
4. **True:** By the Fundamental Theorem of Calculus, $f'(x) = 3x^2 - 12$.
5. **False:** The roots of $f'(x)$ are $\\pm 2$. $f''(2) = 12$ (positive), meaning $x=2$ is a local *minimum*, not a maximum.

```{r solutionlist, echo = FALSE, results = "asis"}
exams::answerlist(c(
  "True",
  "True",
  "False",
  "True",
  "False"
), markup = "markdown")
```

Meta-information
================
extype: mchoice
exsolution: 11010
exname: Multiple Choice Math Capstone
exshuffle: TRUE
expoints: 3
