I think there are no two points on the curve that share the same tangent (i.e the line is a tangent to two points on the curve), however two points may share the same gradient and hence a common tangent line and there are infinitely many of these.
There are two points on the curve y = x^4 - 2x^2 - x that have a common tangent line. Find those points.
So far I think I know what needs to be found but I cannot put the pieces together.
y-prime = 4x^3 - 4x - 1
i need a y prime that yields a slope such that the equation y=mx+b will intersect the initial curve such that the new x value will give the same derivative value as the first point (thus the same sloped tangent)
but i do not know how to do this
If you have a hard way solution there's no point me spending time posting another one which would probably be equivalent. Perhaps you could take the time to post the details of your hard way solution and I'll let you know if it's the same as mine.
There's an easy way - I posted it - that boils down to intelligent guess work. Obviously the solution for x has to lie within the interval for which f'(x) = constant has more than one solution for x. A graphical approach shows how this interval would be found. x = 1 and x = -1 is an obvious candidate solution to test. insight gets rewarded ......
yes i had posted the easy way i thought... i could have sworn i did make a post and now its not here... oh it must be in the other thread...
i sat down and made an accurate sketch of the graph and stumbled on the easy solution....
here is my teacher's solution: the hard way - let me know what you think
Q: y=x^4 - 2x^2 - x has two points sharing the same tangent. Find those points.
ANS: let the points be (b,f(b)) and (a,f(a))
let b < a
y prime = 4x^3 - 4x -1
so the slope of the tangent is
4a^3 - 4a -1 (equation 1) OR 4b^3 - 4b -1
4a^3 - 4a -1 = 4b^3 - 4b -1
a^3 - b^3 = a - b (after simplifying)
(a-b)(a^2 + ab + b^2) = a-b
a^2 + ab + b^2 = 1 (equation 2)
the slope of the tangent is also [f(a) - f(b)] / a-b
[a^4 - 2a^2 - a - (b^4 - 2b^2 - b) ] / a-b
= [a^4 - b^4 - 2(a^2-b^2) - (a-b)] / a-b
= [(a-b)(a^3 + a^2b + ab^2 + b^3) - 2(a-b)(a+b) - (a-b)] / (a-b)
= (a^3 + a^2b + ab^2 + b^3 - 2(a+b) - 1
= a^2(a+b) + b^2(a+b) - 2(a+b) -1
= (a+b) (a^2 + b^2 -2 ) - 1 = y prime = 4a^3 - 4a - 1 (from 1)
from equation 2 (a+b)(1-ab-2) = 4a(a^2 -1)
(a+b)(-1-ab) = 4a(-ab-b^2) from 2 again
(a+b)(-1-ab) = -4ab(a+b)
there are two cases here either a+b = 0 or a+b is not 0
if a+b is not zero then:
-1-ab = -4ab
3ab = 1 (equation N)
if N is true then substituting into equation 2 gives
a^2 + ab + b^2 = 3ab
a^2 - 2ab + b^2 = 0
(a-b)^2 = 0
a=b
but a is not equal to be as is given in the equation and was stated at the beginning
therefore a+b = 0
so we know a = -b
substitue into equation 2 gives:
(-b)^2 - b^2 + b^2 = 1
b^2 = 1
b = -1
(we began by saying b < a, and since a = -b, we need b < 0 and a > 0)
so b = -1 and a = 1
therefore the points are (-1,0) and (1,-2)
we can verify this by checking the derivative at -1 and they are equal
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i hope i made no typos.... if something doesnt read clearly just ask
Mine is the same (more or less) up to and including the line I've redded.
Continuing from that point:
(a+b)(-1-ab) + 4ab(a+b) = 0
=> (a + b)(3ab - 1) = 0
=> a + b = 0 or 3ab = 1.
From a + b = 0 you get a = 1 and b = -1 or a = -1 and b = 1.
From 3ab = 1 you get or . But b < a => contradiction.
Therefore the solution is x = 1 and x = -1.
Sorry for bumping such an old thread – I was solving a similar problem on another forum when I actually found this thread on Google.
Here’s how I do it.
Let . If the equation of the common tangent is we have
We want to find two distnct points at and such that
and
We will show that . If then ; substituting into gives ; also and so . Hence are roots of the quadratic equation , i.e. . We don’t want this as must be distinct.
Therefore . Substituting into gives , i.e. .