Limits Continuity And Differentiability Question 6
Question: If the polynomial equation $ a _{n}x^{n}+{a _{n-1}}{x^{n-1}}+…..+a_2x^{2}+a_1x+a_0=0 $ ,n positive integer, has two different real roots $ \alpha $ and $ \beta $ , then between $ \alpha and\beta $ , the equation $ na _{n}{x^{n-1}}+(n-1){a _{n-1}}{x^{n-2}}+….+a_1=0 $ has
Options:
A) Exactly one root
B) At most one root
C) At least one root
D) No root
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ f(x)=a _{n}x^{n}+{a _{n-1}}{x^{n-1}}+……+a_2x^{2}+a_1x+a_0 $ Which is a polynomial function in x of degree n.
Hence, f(x) is continuous and differentiable for all x. Let $ \alpha <\beta $ .
We are given, $ f(\alpha )=0=f(\beta ). $
By Rolle’s theorem, $ f’(c)=0 $ for some value c, $ \alpha <c<\beta $ .
Hence, the equation $ f’(x)=na _{n}{x^{n-1}}+(n-1){a _{n-1}}{x^{n-2}}+…+a_1=0 $ has at least one root between $ \alpha $ and $ \beta $ .
BETA
