Maximum
", "Minimum
", "Point of inflection
"], "displayColumns": 0, "distractors": ["", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 1, 0], "marks": 0}], "type": "gapfill", "prompt": "Let $y = \\simplify{{a1}x^2 + {b1}x + {c1}}$.
\n$\\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = $ [[0]]
\nEnter the coordinates of the stationary point of $y$: $\\big($ [[1]] $, $ [[2]] $\\big)$
\n$\\dfrac{\\mathrm{d}^2y}{\\mathrm{d}x^2} = $ [[3]]
\nWhat is the nature of the stationary point of $y$?
\n[[4]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "2*{a2}x+{b2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{sx2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "fractionnumbers", "marks": 1, "vsetrangepoints": 5}, {"answer": "{sy2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "fractionnumbers", "marks": 1, "vsetrangepoints": 5}, {"answer": "2{a2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["Maximum
", "Minimum
", "Point of inflection
"], "displayColumns": 0, "distractors": ["", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0], "marks": 0}], "type": "gapfill", "prompt": "Let $z = \\simplify{{a2}x^2+{b2}x+{c2}}$.
\n$\\dfrac{\\mathrm{d}z}{\\mathrm{d}x} = $ [[0]]
\nEnter the coordinates of the stationary point of $z$: $\\big($ [[1]] $, $ [[2]] $\\big)$
\n$\\dfrac{\\mathrm{d}^2z}{\\mathrm{d}x^2} = $ [[3]]
\nWhat is the nature of the stationary point of $z$?
\n[[4]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "3*{m}*x^2", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "fractionnumbers", "marks": 1, "vsetrangepoints": 5}, {"answer": "{q}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "fractionnumbers", "marks": 1, "vsetrangepoints": 5}, {"answer": "6*{m}*x", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["Maximum
", "Minimum
", "Point of inflection
"], "displayColumns": 0, "distractors": ["", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 0, 1], "marks": 0}], "type": "gapfill", "prompt": "Let $t = \\var{m}x^3+\\var{q}$.
\n$\\dfrac{\\mathrm{d}t}{\\mathrm{d}x} = $ [[0]]
\nEnter the coordinates of the stationary point of $t$: $\\big($ [[1]] $, $ [[2]] $\\big)$
\n$\\dfrac{\\mathrm{d}^2t}{\\mathrm{d}x^2} = $ [[3]]
\nWhat is the nature of the stationary point of $t$?
\n[[4]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "For the following, find the stationary point and determine its nature. For your answers, where appropriate, write your solutions as fractions, NOT decimals, and cancel down where possible.
", "tags": ["ACC1012", "acc1012", "checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "$\\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = \\simplify{{2a1}x+{b1}}$.
\nTo find the $x$-coordinate of the stationary point, solve $\\frac{\\mathrm{d}y}{\\mathrm{d}x} = 0$ for $x$:
\n\\[ \\begin{align} \\frac{\\mathrm{d}y}{\\mathrm{d}x} = \\simplify{{2a1}x + {b1}} &= 0 \\\\ x &= \\simplify[all,fractionNumbers]{{sx1}} \\end{align} \\]
\nFind the $y$-coordinate by substituting this value of $x$ into the definition of $y(x)$:
\n\\[\\begin{align} \\simplify[fractionnumbers]{y({sx1})} &= \\simplify[basic,fractionnumbers]{{a1}{sx1}^2+{b1}{sx1}+{c1}} \\\\ &= \\simplify[fractionnumbers]{{sy1}} \\end{align}\\]
\nFinally, to determine the nature of the stationary point, look at $\\frac{\\mathrm{d}^2y}{\\mathrm{d}x^2}$ at $x = \\simplify[fractionnumbers]{{sx1}}$.
\n\\[ \\frac{\\mathrm{d}^2y}{\\mathrm{d}x^2} = \\simplify{{2*a1}} \\]
\nThis is positive, so the stationary point is a minimum.
\n$\\dfrac{\\mathrm{d}z}{\\mathrm{d}x} = \\simplify{{2a2}x+{b2}}$.
\nTo find the $x$-coordinate of the stationary point, solve $\\frac{\\mathrm{d}z}{\\mathrm{d}x} = 0$ for $x$:
\n\\[ \\begin{align} \\frac{\\mathrm{d}z}{\\mathrm{d}x} = \\simplify{{2a2}x + {b2}} &= 0 \\\\ x &= \\simplify[all,fractionNumbers]{{sx2}} \\end{align} \\]
\nFind the $y$-coordinate by substituting this value of $x$ into the definition of $z(x)$:
\n\\[\\begin{align} \\simplify[fractionnumbers]{z({sx1})} &= \\simplify[basic,fractionnumbers]{{a2}{sx2}^2+{b2}{sx2}+{c2}} \\\\ &= \\simplify[fractionnumbers]{{sy2}} \\end{align}\\]
\nFinally, to determine the nature of the stationary point, look at $\\frac{\\mathrm{d}^2z}{\\mathrm{d}x^2}$ at $x = \\simplify[fractionnumbers]{{sx2}}$.
\n\\[ \\frac{\\mathrm{d}^2z}{\\mathrm{d}x^2} = \\simplify{{2*a2}} \\]
\nThis is negative, so the stationary point is a maximum.
\n$\\dfrac{\\mathrm{d}t}{\\mathrm{d}x} = \\simplify{{3m}x^2}$.
\nTo find the $x$-coordinate of the stationary point, solve $\\frac{\\mathrm{d}t}{\\mathrm{d}x} = 0$ for $x$:
\n\\[ \\begin{align} \\frac{\\mathrm{d}t}{\\mathrm{d}x} = \\simplify{{3m}x^2} &= 0 \\\\ x &= 0 \\end{align} \\]
\nFind the $y$-coordinate by substituting this value of $x$ into the definition of $t(x)$:
\n\\[\\begin{align} z(0) &= \\simplify[basic,fractionnumbers]{{m}*0^3 + {q}} \\\\ &= \\var{q} \\end{align}\\]
\nFinally, to determine the nature of the stationary point, look at $\\frac{\\mathrm{d}^2t}{\\mathrm{d}x^2}$ at $x = 0$.
\n\\[ \\frac{\\mathrm{d}^2t}{\\mathrm{d}x^2} = \\simplify{{6m}x} \\]
\n\\[ \\left.\\frac{\\mathrm{d}^2t}{\\mathrm{d}x^2} \\right\\rvert_{x=0} = \\var{6m} \\times 0 = 0 \\]
\nThis is zero, so the stationary point is a point of inflection.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}