{
  "problem": "$f(x, y) = -x^3+2y^2+x$ What are all the critical points of $f$ ? Choose 2 answers: Choose 2 answers: (Choice A) A $\\left( -\\dfrac{\\sqrt{3}}{3}, 0 \\right)$ (Choice B) B $\\left( \\dfrac{\\sqrt{3}}{3}, 0 \\right)$ (Choice C) C $\\left( -1, 0 \\right)$ (Choice D) D $\\left( 1, 0 \\right)$",
  "hints": [
    "A critical point of a scalar field $f$ is where $\\nabla f = \\bold{0}$. [What's that bolded 0?]",
    "Let's find the gradient of $f$ ! $\\nabla f = \\begin{bmatrix} -3x^2 + 1 \\\\ \\\\ 4y \\end{bmatrix}$ We want each component of the gradient to equal zero, so we want to solve the system of equations below. $\\begin{cases} -3x^2 + 1 = 0 \\\\ \\\\ 4y = 0 \\end{cases}$",
    "The second equation forces $y = 0$. We can rewrite the first equation to say: $x^2 = \\dfrac{1}{3}$ Solving, we get $x = \\pm \\dfrac{\\sqrt{3}}{3}$.",
    "Therefore, $f$ has critical points at: $\\begin{aligned} &\\left( \\dfrac{\\sqrt{3}}{3}, 0 \\right) \\\\ \\\\ &\\left( -\\dfrac{\\sqrt{3}}{3}, 0 \\right) \\end{aligned}$"
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/compute-critical-points/16072430873.json'}