A cubic with a maximum and minimum point is given. Question is to determine coordinates of the minimum and maximum point.
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\nAbove is the graph of some function $f$.
\nWhat are the coordinates of its maximum point? [[0]]
\nWhat are the coordinates of its minimum point? [[1]]
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", "advice": "(i) A maximum point is a point where regardless if you move right or left, the height will decrease. A visual analogy would be a hill: if you're at the top of a hill, no matter which direction you go your height will decrease. So you're looking for a part of the graph which is 'like a hill', and in this graph the point is at $(\\var{xmax}, \\var{ymax})$.
\n(ii) A minimum point is the opposite of a maximum point (or an upside-down version of a maximum point, if you like). The analogy in this case would be a valley: no matter which direction you go your height will increase. In this graph, the minimum point is at $(\\var{xmin}, \\var{ymin})$.
", "variables": {"ymin": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "ymin", "definition": "a*(xmin)^3+b*(xmin)^2+c*(xmin) + d"}, "xmin": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "xmin", "definition": "if(a=1, max(r1/3, r2/a), min(r1/3, r2/a))"}, "a": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "definition": "random(-1..1 except 0)"}, "xmax": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "xmax", "definition": "if(a=1, min(r1/3, r2/a), max(r1/3, r2/a))"}, "r1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "definition": "random(-1..1)*3"}, "c": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "definition": "r1*r2"}, "r2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "definition": "random(-1..1 except 1/2*(a*r1/3 + mod(a*r1,2)))*2-mod(a*r1,2)"}, "d": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "definition": "random(-2..2)"}, "ymax": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "ymax", "definition": "a*(xmax)^3+b*(xmax)^2+c*(xmax) + d"}, "b": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "definition": "-(r1*a + 3*r2)/2"}}, "functions": {"plotgraph": {"definition": "// This functions plots a cubic with coefficients a,b,c,d\n// It creates the board, sets it up, then returns an\n// HTML div tag containing the board.\n\n\n// Max and min x and y values for the axis.\nvar x_min = -6;\nvar x_max = 6;\nvar y_min = -10;\nvar y_max = 10;\n\n\n// First, make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n '500px',\n '600px',\n {\n boundingBox: [x_min,y_max,x_max,y_min],\n axis: false,\n showNavigation: true,\n grid: true\n }\n);\n\n\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\n\n\n// Plot the function.\n board.create('functiongraph',\n [function(x){ return a*x*x*x+b*x*x+c*x + d;},x_min,x_max]);\n\n\n\n\nreturn div;", "type": "html", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["d", "number"]], "language": "javascript"}}, "preamble": {"js": "", "css": ""}, "type": "question", "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}]}]}], "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}]}