2a:null 2b:["$","title",null,{"children":"Function optimization | Mattias Villani | Observable"}] 31:T445,Plot.plot({ title: latexfunc, color: { legend: true, domain: plotlabels.labels, range: plotlabels.colors }, x: { label: "x", axis: true, domain: xdomain }, y: { label: "f(x)", axis: true, domain: ydomain }, grid: inputs.showgridlines, marks: [ //Plot.ruleY([0]), Plot.ruleX(zeroderivpoints, {filter: inputs.plotderivzero, stroke: colors[2]}), Plot.lineY(funcdata, {x: "x", y: "f", stroke: colors[0], strokeWidth: 3}), Plot.line([[x0 - 0.2*xrange, tangentline(x0 - 0.2*xrange)], [x0 + 0.2*xrange, tangentline(x0 + 0.2*xrange)] ], {stroke: colors[1], strokeWidth: 3}), Plot.dot([[x0, f(x0)]], {fill: "black", r:4}), Plot.text([{x: xmidpoint, y: ydomain[1] - 0.1*(ydomain[1]-ydomain[0])}], {x: "x", y: "y", fontSize: 16, dy: -45, textAnchor: "left", text: [`f(x) = ${f(x0).toFixed(3)} f'(x) = ${derivative(x0).toFixed(3)} f''(x) = ${derivative2(x0).toFixed(3)}`], fill: "black"}), ] })32:[] 2e:{"id":"b165fc8da150ec34","slug":"function-optimization","title":"Function optimization","thumbnail":"a377652c5c520804f88871679058354019f96ac39b7ff59aac99ae163119a1ec","access_level":"public","create_time":"$D2025-03-13T18:34:00.297Z","update_time":"$D2025-03-25T12:28:06.171Z","view_time":null,"first_public_time":"$D2025-03-13T21:38:18.638Z","fork":{"id":"32fd1a513378ecaa","slug":"some-functions-and-their-derivatives","title":"A function and its derivatives","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":747,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"186f48bbffc13b97","slug":"stm","title":"STM","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Function optimization","cells":[{"id":0,"value":"# Function optimization","pinned":false,"hidden":false,"mode":"md"},{"id":185,"value":"> *This notebook explains how to find the maximum or minimum of a function using ideas from calculus.*","pinned":false,"hidden":false,"mode":"md"},{"id":397,"value":"The **maximum value** of a function ${tex`f(x)`} is the largest value of the function over all ${tex`x`} in the domain ${tex`\\mathcal{X}`}. The value of the input that maximizes ${tex`f(x)`} is called the **argument of the maximum**, or simply the **maximizer**, and is denoted by ${tex`x_\\mathrm{max}`}. We write this symbolically as \n${tex.block`x_\\mathrm{max} = \\argmax_{x\\in \\mathcal{X}} f(x)`}\n\nIf the maximizer of ${tex`f(x)`} is not at the boundary of ${tex`\\mathcal{X}`}, then candidate maximizers ${tex`x_\\mathrm{max}`} can be found by solving the equation\n${tex.block`f'(x_\\mathrm{max})=0,`}\nprovided that the derivative exists. This is called **the first derivative test**.\n\nTo see why a maximizer has zero derivative, try the interactive plot below with the square function ${tex`f(x)=1 - x^2`}. Move the evaluation point and check that the highest point of the curve is where the derivative is zero, i.e. where the yellow tangent line is horizontal.","pinned":false,"hidden":false,"mode":"md"},{"id":636,"value":">

**Example**. We can find the maximum of the square function ${tex`f(x)=1 - x^2`} by calculating its first derivative ${tex`f'(x)=-2x`} and solving ${tex`f'(x)=-2x=0`}, which gives the maximizer ${tex`x_\\mathrm{max}=0`}.","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"viewof funcname = Inputs.select([\"square\", \"sin\", \"dampedsin\", \"poly 3rd degree\"], {value: \"square\", label: \"Function f(x)\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":441,"value":"viewof inputs = Inputs.form({\n x0: Inputs.range(xdomain, {value: -1, step: .01, label:\"x-value\"}),\n plotderivzero: Inputs.toggle({label: \"plot zero deriv points\", value: false}),\n showgridlines: Inputs.toggle({label: \"show gridlines\", value: true})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":143,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":601,"value":"Unfortunately, not all candidate maximizers that satisfy the equation ${tex.block`f'(x_\\mathrm{max})=0`} give the true **global** maximum. The square function above is *concave* so there is only one candidate, and all is fine. But in general there can be two types of false candidates:\n\n+ *minimizers*\n+ *local maximizers* \n\n","pinned":false,"hidden":false,"mode":"md"},{"id":642,"value":">

**Example**. Let's find the maximum of the third degree polynomial function ${tex`f(x) = 1 + 2x^2 -0.5x^3`} over the domain ${tex`[-5,5]`}. We do the first derivative test and solve ${tex`f'(x)=(4 - 1.5 x) x = 0`}, which has two solutions ${tex`x_1=0`} and ${tex`x_2=8/3 \\approx 2.666`}. Try selecting the function '*poly 3rd degree*' in the interactive plot above. Note that the first solution is a local minimum and the second solution is a local maximum.","pinned":false,"hidden":false,"mode":"md"},{"id":648,"value":"The **second derivative test** can be used check if a candidate ${tex`x_\\text{cand}`} is a minimizer or a maximizer:\n- if ${tex`f''(x_\\text{cand})<0`} then ${tex`x_\\text{cand}`} is a maximizer\n- if ${tex`f''(x_\\text{cand})>0`} then ${tex`x_\\text{cand}`} is a minimizer\n- if ${tex`f''(x_\\text{cand})=0`} then the test is inconclusive","pinned":false,"hidden":false,"mode":"md"},{"id":687,"value":">

**Example**. The second derivate of the third degree polynomial function ${tex`f(x) = 1 + 2x^2 -0.5x^3`} in the previous example is ${tex`f''(x) = 4 - 3x`}. Since ${tex`f''(0) = 4`} is positive, we know that ${tex`x_1=0`} is a local minimizer. The second candidate is a local maximizer since ${tex`f''(8/3) = -4`} is negative.","pinned":false,"hidden":false,"mode":"md"},{"id":692,"value":"A candidate maximizer can determined to be a global maximizer by comparing its function value ${tex`f(x_{\\mathrm{cand}})`} to all the other candidates' function values.\n\nNote also that the global maximizer may occur at one of the boundaries of the domain ${tex`\\mathcal{X}=[a,b]`}, so we need also need to check if ${tex`f(a)`} or ${tex`f(b)`} are larger than the candidate maximizer found by the first derivative test.","pinned":false,"hidden":false,"mode":"md"},{"id":723,"value":">

**Example**. The global maximum of the third degree polynomial function ${tex`f(x) = 1 + 2x^2 -0.5x^3`} over the domain ${tex`[-5,5]`} actually occurs at the boundary since ${tex`f(-5) = 1 + 2(-5)^2 -0.5(-5)^3=113.5`} is larger than the local maxima obtained above where ${tex`f(8/3)=12.85185`}","pinned":false,"hidden":false,"mode":"md"},{"id":31,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":28,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":96,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":166,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":16,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":19,"value":"latexfunc = {\n if (funcname == \"square\"){\n return tex`f(x) = 1 - x^2`\n }\n if(funcname == \"dampedsine\"){\n return tex`f(x) = \\sin(3x)\\exp(-x)`\n }\n if(funcname == \"poly 3rd degree\"){\n return tex`f(x) = 1 + 2x^2 - 0.5x^3` \n }\n \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":576,"value":"x0 = inputs.x0","pinned":false,"hidden":false,"mode":"ojs"},{"id":439,"value":"h = 0.01","pinned":false,"hidden":false,"mode":"ojs"},{"id":21,"value":"f = {\n if (funcname == \"square\"){\n return function(x){return 1 - (x**2)}\n }\n if(funcname == \"sin\"){\n return function(x){return Math.sin(x)}\n }\n if(funcname == \"dampedsin\"){\n return function(x){return Math.sin(3*x)*Math.exp(-x)}\n }\n if(funcname == \"poly 3rd degree\"){\n return function(x){return 1 + 2*x**2 - 0.5*x**3}\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":23,"value":"derivative = {\n if (funcname == \"square\"){\n return x => -2*x \n }\n if(funcname == \"sin\"){\n return x => Math.cos(x)\n }\n if(funcname == \"dampedsin\"){\n return x => Math.exp(-x)*(3*Math.cos(3*x) - Math.sin(3*x))\n }\n if(funcname == \"poly 3rd degree\"){\n return x => 4*x - (3/2)*x**2\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":25,"value":"derivative2 = {\n if (funcname == \"square\"){\n return x => -2 \n }\n if(funcname == \"sin\"){\n return x => -Math.sin(x)\n }\n if(funcname == \"dampedsin\"){\n return x => -2*Math.exp(-x)*(3*Math.cos(3*x) + 4*Math.sin(3*x))\n }\n if(funcname == \"poly 3rd degree\"){\n return x => 4 - 3*x\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":546,"value":"zeroderivpoints = {\n if (funcname == 'dampedsin'){\n return [(2/3)*Math.atan((1/3)*(-1 - Math.sqrt(10))), \n (2/3)*(Math.PI +Math.atan((1/3)*(-1 - Math.sqrt(10)))),\n -(2/3)*(Math.PI - Math.atan((1/3)*(Math.sqrt(10) - 1))),\n (2/3)*Math.atan((1/3)*(Math.sqrt(10) - 1))\n ]\n }\n if(funcname == \"sin\"){\n return [-(3/2)*Math.PI, -Math.PI/2, Math.PI/2, (3/2)*Math.PI]\n }\n if(funcname == \"square\"){\n return [0]\n }\n if(funcname == \"poly 3rd degree\"){\n return [0,8/3]\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":437,"value":"tangentline = function(x){\n const slope = (f(x0+h)-f(x0))/h\n const b = f(x0)-slope*x0\n return slope*x+b\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":36,"value":"xdomain = {\n if (funcname == \"square\"){\n return [-3, 3]\n }\n if (funcname == \"sin\"){\n return [-2*Math.PI, 2*Math.PI]\n }\n if(funcname == \"poly 3rd degree\"){\n return [-5,5]\n }\n if (funcname == \"dampedsin\"){\n return [-2,2]\n }else{\n return [-5,5]\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":481,"value":"xrange = xdomain[1] - xdomain[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":535,"value":"xmidpoint = d3.mean(xdomain)","pinned":false,"hidden":false,"mode":"ojs"},{"id":145,"value":"funcdata = {\n const ngrid = 500;\n let funcval = [];\n const xgrid = d3.range(xdomain[0], xdomain[1], (xdomain[1] - xdomain[0])/ngrid)\n for (let i = 0; i < ngrid; ++i) {\n funcval.push({x: xgrid[i], f: f(xgrid[i]), fprime: derivative(xgrid[i])})\n }\n return funcval\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":173,"value":"plotlabels = {\n let labels = [\"f(x)\", \"tangent line\"];\n let labelColors = [colors[0], colors[1]]\n if (inputs.plotderivzero){\n labels.push(\"zero derivative\")\n labelColors.push(colors[2])\n }\n return {labels: labels, colors: labelColors}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":202,"value":"ydomain = {\n let ylimits = d3.extent(funcdata.map(d => d.f))\n const yrange = ylimits[1] - ylimits[0]\n ylimits[0] = ylimits[0] - 0.2*yrange\n ylimits[1] = ylimits[1] + 0.2*yrange\n return ylimits\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":392,"value":"tiplabels = function(d, add_derivatives){\n let labels = [`x=${d.x.toFixed(3)}`, `f(x)=${d.f.toFixed(3)}`]\n if (add_derivatives[0]){labels.push(`f'(x)=${d.fprime.toFixed(3)}`)}\n return labels\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":653,"value":"polyfunc = function(x){\n return 1 + 2*x**2 -0.5*x**3\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":656,"value":"polyfuncdata = d3.range(-5, 5, 0.01).map(d => ({x: d, f: polyfunc(d)}))","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}} 2f:"$2e:body"