# Unique Line Tangent to 2 points on the curve y=x^4+x^3

jonbenedick shared this question 7 years ago

How do you draw a unique line tangent to the two points on the curve y=x^4+x^3 1

Do you mean like this, or something else?

http://tube.geogebra.org/ma... 1

hello

i got with Maxima that you must use the point A=((sqrt(3)-1)/4,0) and f(x)=x^4+x^3 then Tangent[A, f]

i can not get it with CAS because i do not know how to say to CAS a!=c for solutions of equations

saludos  1

hello

it is possible with CAS substituting by hand for avoiding 0/0

saludos  1

The common tangent slope is the same of the tangent at point A(-1/4,f(-1/4)), being -1/4 the solution of equation: third derivative of f(x)=0. See att'd figure.  1

Finalized. Files: ct.jpg 1

More details.  1

MatLab solution, according to mathmagic's scheme.

% ------------- twovars2.m ------------- (May.20,2015) ------

% ----- Successive approx. 2x2 system of NON LINEAR equations ------

% --------------------------------------------------------------------------

clc, clear, close all % clear screen

format long

% -------------------------------------------------

% 2 starting values of iteration

a=0.8; c=-1.2; % a= right-hand abscissa; c= left-hand abscissa

%

% System to solve: F1(a,c)=0; F2(a,c)=0

while 1 % Start iteration

F(1)= (4*a^3 + 3*a^2)*(a-c) - (a^4+a^3-c^4-c^3); % F1

F(2)= (4*c^3 + 3*c^2)*(a-c) - (a^4+a^3-c^4-c^3); % F2

% ------- W = jacobian matrix ----------

W(1,1)= (12*a^2 + 6*a)*(a-c)+ (4*a^3 + 3*a^2) -4*a^3-3*a^2; % F11= dF1/da |F12 F12|

W(1,2)= -(4*a^3 + 3*a^2)+4*c^3+3*c^2; % F12= dF1/dc W= | |

% |F21 F22|

W(2,1)= 4*c^3 + 3*c^2-4*a^3-3*a^2; % F21= dF2/da

W(2,2)= (12*c^2+6*c)*(a-c)-(4*c^3 + 3*c^2) +4*c^3+3*c^2; % F22= dF2/dc

%

P=inv(W)*F'; % Xi+1= Xi - inv(W)*F(Xi); iteration vars Xi=(ai,ci)

%

a1 = a - P(1); c1 = c - P(2);

fprintf ('\n ai = %9.6f ci = %9.6f', a1,c1);

if abs(P(1)) < 1e-8 & abs(P(2)) < 1e-8

break;

end

a = a1; c = c1;

end; % end while 1 (iteration)

%

fprintf ('\n\n RESULTS \n a = %9.6f c = %9.6f \n', a1,c1);

%

% ------- EOF: twovars2.m ---------

% Output, convergence per successive approximations

% ai = 0.559666 ci = -0.992490

% ai = 0.388581 ci = -0.847549

% ai = 0.274385 ci = -0.753663

% ai = 0.209812 ci = -0.702834

% ai = 0.186168 ci = -0.685221

% ai = 0.183063 ci = -0.683046

% ai = 0.183013 ci = -0.683013

% ai = 0.183013 ci = -0.683013

% ai = 0.183013 ci = -0.683013

%

% RESULTS

% a = 0.183013 c = -0.683013