// Numbas version: exam_results_page_options {"name": "Test grade 8 - 13 (English)", "duration": 5400, "metadata": {"description": "

Here you will find task for the 8. to 13. grade.

", "licence": "All rights reserved"}, "allQuestions": true, "shuffleQuestions": false, "percentPass": "50", "timing": {"allowPause": true, "timeout": {"action": "warn", "message": "

Time is up. You have to submit.

"}, "timedwarning": {"action": "warn", "message": "

5 minutes left!

"}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "warnifunattempted", "message": "

Some of the questions are unanswered, do you still want to submit your answer?

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After finishing your test, you have the oppurtunity to review your answers.

", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true}, "type": "exam", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Fractions 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Robert F\u00f8rland", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/827/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "": []}, "parts": [{"stepsPenalty": "0", "prompt": "

Calculate

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$\\displaystyle\\simplify{(x+{a})/({b}x)-({c}x-{d})/({ee}x)}=$[[0]]

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", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Expand the fractions to get a common denominator. Then add and subract.

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(Simplify if possible.)

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({ee-b*c}x+{ee*a+b*d})/({b*ee}x)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": "0", "prompt": "

Calculate

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$\\displaystyle{\\frac{\\frac{x}{\\var{j}}+\\frac{1}{\\var{l}}}{\\frac{\\var{g}x}{\\var{j*l}}+\\frac{1}{\\var{j}}}} =$[[0]]

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", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Find the common denominator for all the small fractions. Then multiply by this on both sides of the main fraction line.

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Example: 

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\\[\\frac{\\frac{2x}{3}+\\frac{1}{5}}{\\frac{4x}{15}-\\frac{1}{3}}=\\frac{(\\frac{2x}{3}+\\frac{1}{5})\\cdot 15}{(\\frac{4x}{15}-\\frac{1}{3})\\cdot 15}=\\frac{2x\\cdot 5+1\\cdot3}{4x-1\\cdot 5}=\\frac{10x+3}{4x-5}\\]

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\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({l}x+{j})/({g}x+{l})", "marks": "1", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

Calculate and give your answer as a fraction or an algebra expression. Use  /  as the fraction line, for example $\\frac{x-2}{3}$ is written as (x-2)/3 and $\\frac{(x+1)}{(x-4)}$ as (x+1)/(x-4)

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Add, subtract, multiply and divide algebraic fractions.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Squared and cubed expressions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Robert F\u00f8rland", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/827/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "": []}, "parts": [{"stepsPenalty": 0, "prompt": "

Calculate

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$\\displaystyle{\\left(\\frac{y}{\\var{k}}\\right)^\\var{m+1}} =$[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

$(a\\cdot b)^n = a^n\\cdot b^n$

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$\\displaystyle{\\left(\\frac{a}{b}\\right)^n=\\frac{a^n}{b^n}}$

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Use this video for further help:

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", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "y^{m+1}/{k}^{m+1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": "0", "prompt": "

Calculate

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$\\displaystyle{\\left(\\var{a}x\\right)^\\var{n}} =$[[0]]

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", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

$(a\\cdot b)^n = a^n\\cdot b^n$

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$\\displaystyle{\\left(\\frac{a}{b}\\right)^n=\\frac{a^n}{b^n}}$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}^{n}*x^{n}", "marks": "1", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Simplify:

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$\\displaystyle{(\\var{b}y)^\\var{-m+1}\\cdot (-\\var{a}y)^\\var{k}} =$[[0]]

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", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

$a^n\\cdot a^m=a^{n+m} $

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$\\displaystyle{\\frac{a^n}{a^m}=a^{n-m}}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{b}^{-m+1}*{-a}^{k}*y^{-m+k+1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Calculate:

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$\\displaystyle{\\left(\\var{d}\\cdot a^2\\right)^\\var{l}} = $[[0]] 

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

$a^n\\cdot a^m=a^{n+m} $

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$\\displaystyle{\\frac{a^n}{a^m}=a^{n-m}}$

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$\\displaystyle{\\left(a^n\\right)^m=a^{n\\cdot m}}$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{d}^{l}*a^{2*l}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Simplify:

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$\\displaystyle{\\frac{\\left(y^\\var{n}z\\right)^{-\\var{m}}\\cdot \\left(y z^\\var{k}\\right)^\\var{m}}{\\left(y^{-\\var{n-1}}z\\right)^\\var{m+1}}} = $[[0]] 

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

$a^n\\cdot a^m=a^{n+m} $

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$\\displaystyle{\\frac{a^n}{a^m}=a^{n-m}}$

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$\\displaystyle{\\left(a^n\\right)^m=a^{n\\cdot m}}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "y^{n-1}*z^{-2*m+k*m-1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

If you are going to write for example $3^{-2}\\cdot x^7$, you have to write it like this 3^(-2)*x^7. 

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Another example: $x^2\\cdot y$ has to be written as x^2*y

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Working with powers

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Growth rate - by graph and calculating", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/rett_linje_stigningstall.png", "/srv/numbas/media/question-resources/rett_linje_stigningstall.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Robert F\u00f8rland", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/827/"}], "functions": {"eqnline": {"definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n \n// The curve is described by the equation \n// y = x^2-(a+b)x+a*b\n\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n{boundingBox: [-6,10,6,-10],\n axis: false,\n showNavigation: false,\n grid: true\n});\n \n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,1],{\n drawLabels: true,\n label: {offset: [-4, -10]},\n minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,1],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n// Draw the function\n\n\nboard.create('functiongraph',[function(x){ return (x-a)*(x-b);},-6,6]);\nboard.create('functiongraph',[function(x){ return (2*x1-a-b)*(x-x1)+y1; strokeColor: 'black'},-6,6]);\nboard.create('point',[x2,y2],{size:2});\n\nreturn div;", "type": "html", "language": "javascript", "parameters": [["a", "number"], ["b", "number"], ["x1", "number"], ["y1", "number"], ["x2", "number"], ["y2", "number"]]}}, "ungrouped_variables": ["a", "b", "c", "x1", "y1", "dist", "x2", "y2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

We know that the graph crosses the $x$-axis at both $(\\var{a},0)$ and $(\\var{b},0)$. Since this is a quadratic, we know our equations has two roots, and by the previous observation, they are at $\\var{a}$ and $\\var{b}$. Hence we can write our equation as $\\simplify{y=(x-{a})(x-{b})}$ which simplifies to $\\simplify{y=x^2-({a}+{b})x+({a}*{b})}$.

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To find the coefficients of the turning point of the quadratic, we know the x-coordinate of the turning point will correspond to the solution to $dy/dx=0$. So we get $\\simplify{2x-({a}+{b})}=0$ hence $\\simplify{x=({a}+{b})/2}$. We substitute this value of x back into the equation of the quadratic to find the corresponding y-coordinate.

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

Find the growth rate graphically in the point $(\\var{x1}, \\var{y1})$

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The growth rate is: [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Hint: The growth rate to the function in the point $(\\var{x1}, \\var{y1})$ is the slope to the tangent in the same point. This slope kan be found by reading the graph. Just watch the following example:

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Here the slope would be $a = \\frac{\\Delta x}{\\Delta y}=\\frac{2}{1}=2$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "2*x1-(a+b)", "minValue": "2*x1-(a+b)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Calculate the growth rate to function $f$ i point A with the coordinates $(\\var{x2}, \\var{y2})$

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Growth rate: [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Hint: The growth rate in the point $(\\var{x2}, \\var{y2})$ is given by $f'(\\var{x2})$.

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "2*x2-(a+b)", "minValue": "2*x2-(a+b)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

{eqnline(a,b,x1,y1,x2,y2)}

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This is the graph for the following function

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$\\displaystyle\\simplify[all, !Noleadingminus]{f(x)=x^2-{a+b}x+{a*b}}$

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with a tangent point in the following coordinates $(\\var{x1}, \\var{y1})$ and point A at the following coordinates $(\\var{x2}, \\var{y2})$

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-3..3 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-2..2 except [0,a,-a])", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "dist": {"definition": "random(-2..2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "dist", "description": ""}, "x1": {"definition": "floor((a+b)/2)+dist", "templateType": "anything", "group": "Ungrouped variables", "name": "x1", "description": ""}, "y1": {"definition": "(x1-a)*(x1-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "y1", "description": ""}, "x2": {"definition": "floor((a+b)/2)-dist", "templateType": "anything", "group": "Ungrouped variables", "name": "x2", "description": ""}, "y2": {"definition": "(x2-a)*(x2-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "y2", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Stationary Points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Robert F\u00f8rland", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/827/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "r1", "r2", "an", "d", "ap", "xmin", "xmax"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

You will find advice for each task in the test.

", "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepsPenalty": 0, "prompt": "

$f(x)=\\simplify[all,!collectNumbers,!noleadingminus, fractionNumbers]{{a}x^3+{b}x^2-{c}x+{d}}$

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$f'(x)=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use derivation rule $(x^n )'=n x^{n-1}$

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Use this link for more information.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "3*{a}x^2+2*{b}x-{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Determine where  $f'(x)$ has its maximum- and minimumpoint.

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Coordinate for maximumpoint: ( [[0]] , [[2]] )

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Coordinate for minimumpoint: ( [[1]] , [[3]] )

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

When vi derivate the function, we get $\\displaystyle f'(x)=\\simplify[std]{{3*a}x^2+{2*b}x-{c}}$.

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To find the stationary points, we have to solve  $\\displaystyle f'(x)=0$ for $x$, 

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$\\simplify[std]{{3*a}x^2+{2*b}x-{c}=0}$.

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This equation has two solutions; $x=\\var{r1}$ og $x=-\\var{r2}$. To classify the stationary points (maximumpoint, minimumpoint or saddle point) we make a sign table for the derivated function.

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To find out more, check out this movie:

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Find the coordinates to the functions stationary points and determine if they are maximum points, minimum point og saddle points.

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Finding the stationary points of a cubic with two turning points

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