Rolle's Theorem

The underlying JavaScript code

board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 10, 7, -6], axis: true});
board.suspendUpdate();
var p = [];
p[0] = board.create('point', [-1,2], {size:3,face:'x',fixed:true});
p[1] = board.create('point', [6,2], {size:3,face:'x',fixed:true});
p[2] = board.create('point', [-0.5,1], {size:2});
p[3] = board.create('point', [2,0.5], {size:2});
var f = JXG.Math.Numerics.lagrangePolynomial(p);
var graph = board.create('functiongraph', [f, -10, 10]);

var r = board.create('glider', [function() { return JXG.Math.Numerics.root(JXG.Math.Numerics.D(f),(p[0].X()+p[1].X())*0.5); },
                                function() { return f(JXG.Math.Numerics.root(JXG.Math.Numerics.D(f),(p[0].X()+p[1].X())*0.5)); },graph], 
                                {name:' ',size:4});
var t = board.create('tangent', [r], {strokeColor:'#ff0000'});
line = board.create('line',[p[0],p[1]],{strokeColor:'#ff0000',dash:1});
board.unsuspendUpdate();