Suppose a cubic polynomial f(x) has three real roots a, b, and c which are not all equal.
Given the slopes of the tangent lines at two of these roots, find the slope of the third.
In other words, if f'(a)=p and f'(b)=q (where p and q are not both 0), find a formula for f'(c) in terms of p and q.
The Formula: f'(c) =  p*q/(p+q)
f(x) = (xa)(xb)(xc)
f(x) = x^3  (a+b+c)x^2 + (ab+ac+bc)x  abc
f'(x) = 3x^2  2(a+b+c)x + (ab+ac+bc)
f'(a) = p = a^2  ab  ac + bc = (ab)(ac)
f'(b) = q = b^2  ab  bc + ac = (ba)(bc)
f'(c) = r = c^2  bc  ac + ab = (ca)(cb)
p+q = f'(a) + f'(b)
p+q = (ab)(ac)  (ab)(bc)
p+q = (ab)(acb+c)
p+q = (ab)^2
p*q = f'(a) * f'(b) = (ab)(ac)(ba)(bc)
p*q = (1) (ab)^2 (ac)(bc)
p*q = (1) (ab)^2 (ca)(cb)
p*q = (1) (ab)^2 * f'(c)
f'(c) =  p*q/(p+q)

Posted by Larry
on 20200922 07:48:35 