# Exercise: Study of Razional Function

Given the function f shown in the grafics view and using GeoGebra commands, described in the previous examples: a. find the domain, X-Axis intersections, Y-Axis intersections, the sign of f, the limits at the endpoints of the domain and the asymptotes; b. find the monotonicity intervals, local and global minima and maxima of f; c. find the convexity and concavity intervals and the inflection points of f; d. draw a qualitative graph of f; e. verify if the graph is correct typing f in the entry bar. Basically, we only require four commands, a function and a GeoGebra tool: the Root and Intersection commands and the function Sgn (to determine the domain, to find the zeros and to study the sign) and the commands Limit and Derivative and finally the instrument Pen. Example protocol No. Name ==> Definition / Command 1 ………………. 2 ………………….. 3 ……………………….. 4 Function g ==> g(x) = sgn(f(x)) 5 Point A ==> (0, f(0)) 6 Point B ==> Root of Numerator[f] 6 …………………………………… 7 Point R ==> Root of Denominator[f] 8 Line a ==> x = x(R) 9 Number LimDx ==> LimitAbove[f, x(R)] 10 Number LimSx ==> LimitBelow[f, x(R)] 11 PenStroke ==> 12 Number LimInf ==> Limit[f, ∞] 13 Number m ==> Limit[f(x) / x, ∞] 14 Number q ==> Limit[f(x) - m x, ∞] 15 Line b ==> y = m x + q 16 PenStroke 17 Function f' ==> f'(x) = f'(x) 18 Point D ==> Root[numerator[f’(x)]] 18 Point E ==> Root[numerator[f’(x)]] 19 Function h ==> h(x) = sgn(f'(x)) 20 Point F ==> (x(D), f(x(D))) 21 Point G ==> (x(E), f(x(E))) 22 Function f'' ==> f''(x) = f''(x) 23 Function p ==> p(x) = sgn(f''(x)) 24 PenStroke 25 Function r ==> f