A) At least one root in [0, 1]
B) At least one root in [1, 2]
C) At least one root in \[[-1,\,0]\]
D) None of these
Correct Answer: A
Solution :
Let \[{f}'(x)\] denotes the quadratic expression \[f'(x)\equiv 3a{{x}^{2}}+2bx+c\], whose antiderivative be denoted by \[f(x)=a{{x}^{3}}+b{{x}^{2}}+cx\] Now \[f(x)\] being a polynomial in \[R,f(x)\] is continuous and differentiable on R. To apply Rolle's theorem. We observe that \[f(0)=0\]and \[f(1)=a+b+c=0,\] by hypothesis. So there must exist at least one value of x, say \[x=\alpha \in (0,1)\] such that \[{f}'(\alpha )=0\,\,\,\Leftrightarrow \,\,\,3a{{\alpha }^{2}}+2b\alpha +c=0\] That is, \[{f}'(x)=3a{{x}^{2}}+2bx+c=0\]has at least one root in [0, 1].
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