(A) Let $f(x) = \frac{a_{3} x^{4}}{4} + \frac{a_{2} x^{3}}{3} + \frac{a_{1} x^{2}}{2} + a_{0} x$. $f(0) = 0$. $f(1) = \frac{a_{3}}{4} + \frac{a_{2}}{3

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(A) Let $f(x) = \frac{a_{3} x^{4}}{4} + \frac{a_{2} x^{3}}{3} + \frac{a_{1} x^{2}}{2} + a_{0} x$. $f(0) = 0$. $f(1) = \frac{a_{3}}{4} + \frac{a_{2}}{3} + \frac{a_{1}}{2} + a_{0} = 0$ (given). Since $f(0) = f(1) = 0$, by Rolle's Theorem, there exists at least one $c \in (0, 1)$ such that $f'(c) = 0$. $f'(x) = a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}$. Thus, the equation $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} = 0$ has at least one real root in the interval $[0, 1]$.

Class 12 Mathematics Continuity and Differentiation Rolle’s theorem, Lagrange's mean value theorem

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For all real values of $a_{0}, a_{1}, a_{2}, a_{3}$ satisfying $a_{0}+\frac{a_{1}}{2}+\frac{a_{2}}{3}+\frac{a_{3}}{4}=0$, the equation $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}=0$ has a real root in the interval

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$[0, 1]$
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$[-1, 0]$
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$[1, 2]$
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$[-2, -1]$

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Rolle’s theorem, Lagrange's mean value theorem

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For all real values of $a_{0}, a_{1}, a_{2}, a_{3}$ satisfying $a_{0}+\frac{a_{1}}{2}+\frac{a_{2}}{3}+\frac{a_{3}}{4}=0$, the equation $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}=0$ has a real root in the interval

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