Problema Solution
Given that f(x) is a fourth degree polynomial function with real coefficients, prove the following statements when f(x)=0:
(1) If a and b are different real solutions of f(x)=0, then f'(x)=0 has at least one solution between a and b.
(2) If f(x)=0 has 4 different real solutions, then f'(x)=0 has 3 different real solutions.
Answer provided by our tutors
(1) By The Mean Value Theorem (or its simpler version of Rolle's Theorem) there is a point c between a and b for which f'(c)=(f(b)-f(a))/(b-a)
Since f(a)=f(b)=0, we have
f'(c)=(0-0)/(b-a)=0
c is a solution for f'(x)=0 between a and b
(2) If f(x)=0 has 4 different solutions a1
a1
and f'(c1)=f'(c2)=f'(c3)=0
In particular c1,c2,c3 are all different numbers, so they consist of 3 different solutions to f'(x)=0