Applications Of Derivatives Question 257
Question: If $ 2a+3b+6c=0, $ then at least one root of the equation $ ax^{2}+bx+c=0 $ lies in the interval
Options:
A) (0, 1)
B) (1, 3)
C) (2, 3)
D) (1, 3)
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Let $ f’(x)=ax^{2}+bx+c $
$ \Rightarrow f(x)=\frac{ax^{3}}{3}+\frac{bx^{2}}{2}+cx+d $
$ \Rightarrow f(x)=\frac{2ax^{3}+3bx^{2}+6cx+6d}{6} $
$ \therefore f(1)=\frac{2a+3b+6c+6d}{6}=\frac{6d}{6}=d $
$ (\therefore 2a+3b+6c=0) $ And $ f(0)=d $ So from Rolle’s Theorem there exists at least one a in (0, 1).
For which $ f’(x)=0. $ Or there is at least one root of $ ax^{2}+bx+c=0 $ in (0, 1).
BETA
