how many tangents to the curve $$y=x^3-3x$$can be drawn from different points in the plane?

Question

anonymous · Answer

$$\large \color{green}{y=x^3-3x}$$

experimentx · Answer

do the usual stuff .. take derivatives, find the critical points.

anonymous · Answer

the first derivative $$y'=0\3x^2-3=0\x=1,x=-1$$

experimentx · Answer

3x^2 - 3 = 0

anonymous · Answer

$$f(1)=-2,f(-1)=2$$

anonymous · Answer

but the didnt ask us to find critical points

anonymous · Answer

if we say there is a fixed $$x=x_0$$ then the tangent line at this point  is
$$y-y_0=(3x_0^2-3)(x-x_0 )\y=-x_0^3+3x+3xx_0^2-3x_0^3-3x+3x_0\y=3x(x_0^2)+x_0(3-4x_0^2)$$

anonymous · Answer

@experimentX check the updated question

anonymous · Answer

the question was not complete...sry

experimentx · Answer

oh ... i thought you might be looking for tangent parallel to x-axis.

anonymous · Answer

but do u understand the new version of the problem

experimentx · Answer

no I don't ... there are infinite number of tangents ... isn't there some constraint?

anonymous · Answer

the concept of  from different points is illusional

experimentx · Answer

?? what does that mean?

anonymous · Answer

i am trying to understand it...the least i am getting is that the tangents are drawn from this points and we cant use the same point twice|dw:1384525771462:dw|

experimentx · Answer

the is only one tangent at one point ... the all tangents are unique.

anonymous · Answer

yes but now we are interste with other points on the graph ...all the points

anonymous · Answer

you can follow the question here also... http://math.stackexchange.com/questions/568214/how-many-tangents-to-the-curve-y-x3-px-can-be-drawn-from-different-points-i

experimentx · Answer

did you mean that tangent passing through any particular point (a,b) ??

anonymous · Answer

yes i mean thats how i wud say i understand it...the problem is not giving any such restrictionss....

experimentx · Answer

|dw:1384527166477:dw|