Assertion (A) and Reason (R) (A) Both A and R are true and R is the correct explanation of A. (B

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Assertion (A) and Reason (R) (A) Both A and R are true and R...

Assertion (A) and Reason (R) (A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$. (B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$. (C) $\mathrm{A}$ is true, $\mathrm{R}$ is false. (D) $\mathrm{A}$ is false, $\mathrm{R}$ is true. Assertion $(\mathbf{A}):$ If $27 a+9 b+3 c+d=0$, then the equation $f(x)=4 a x^{3}+3 b x^{2}+2 c x+d=0$ has at least one real root Iying between $(0,3)$. Reason $(\mathrm{R}):$ If $f(x)$ is continuous in $[a, b]$, derivable in b) such that $f(a)=f(b)$, then ther exists atleast one point (a, $c \in(a, b)$ such that $f^{\prime}(c)=0$

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(\mathbf{A}):$ If $27 a+9 b+3 c+d=0$, then the equation $f(x)=4 a x^{3}+3 b x^{2}+2 c x+d=0$ has at least one real root Iying between $(0,3)$. Reason $(\mathrm{R}):$ If $f(x)$ is continuous in $[a, b]$, derivable in
b) such that $f(a)=f(b)$, then ther exists atleast one point (a, $c \in(a, b)$ such that $f^{\prime}(c)=0$

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Key Concepts

Relationship Between a Function and Its Derivative

In calculus, there is an important connection between a function and its derivative, particularly when the function attains the same value at two distinct points. This relationship implies that between these points there must be at least one location where the rate of change of the function (its derivative) is zero. This principle is not only a direct application of Rolle's theorem but also a critical concept for understanding the behavior of polynomial functions and finding their roots.

Rolle's Theorem

Rolle's theorem is a fundamental result in calculus that states if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if the function has equal values at the endpoints (f(a) = f(b)), then there exists at least one point c in (a, b) at which the derivative f?(c) is zero. This theorem is crucial for establishing the existence of stationary points and lays the groundwork for more general mean value properties of functions.

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