A) Exactly one root
B) At most one root
C) At least one root
D) No root
Correct Answer: C
Solution :
[c] Let \[f(x)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+......+{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}\] 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 \[f'(x)=n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{x}^{n-2}}+...+{{a}_{1}}=0\] has at least one root between \[\alpha \] and \[\beta \].
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