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) = (x-a)(x-b)(x-c)
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 = (a-b)(a-c)
f'(b) = q = b^2 - ab - bc + ac = (b-a)(b-c)
f'(c) = r = c^2 - bc - ac + ab = (c-a)(c-b)
p+q = f'(a) + f'(b)
p+q = (a-b)(a-c) - (a-b)(b-c)
p+q = (a-b)(a-c-b+c)
p+q = (a-b)^2
p*q = f'(a) * f'(b) = (a-b)(a-c)(b-a)(b-c)
p*q = (-1) (a-b)^2 (a-c)(b-c)
p*q = (-1) (a-b)^2 (c-a)(c-b)
p*q = (-1) (a-b)^2 * f'(c)
f'(c) = - p*q/(p+q)
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Posted by Larry
on 2020-09-22 07:48:35 |
Solution
