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. Let $\mathrm{f}(\mathrm{x})$ be a twice differentiable function. Assertion $(\mathrm{A}):$ If $\mathrm{a}<\mathrm{b}<\mathrm{c}<\mathrm{d}$ and $\mathrm{f}(\mathrm{a})=0, \mathrm{f}(\mathrm{b})=1$, $\mathrm{f}(\mathrm{c})=-1, \mathrm{f}(\mathrm{d})=0$, then the minimum number of zeroes $g(x)=\left(f^{\prime}(x)\right)^{2}+f(x) f^{\prime}(x)$ in $[a, d]$ is $4 .$ Reason (R) : If $f(\alpha) f(\beta)<0$ then $f(\gamma)=0$ for some $\alpha<\gamma<\beta$ and if $f(\alpha)=f(\beta)=0$ then $f^{\prime}(\gamma)=0$ for some $\alpha<\gamma<\beta$

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.
Let $\mathrm{f}(\mathrm{x})$ be a twice differentiable function. Assertion $(\mathrm{A}):$ If $\mathrm{a}<\mathrm{b}<\mathrm{c}<\mathrm{d}$ and $\mathrm{f}(\mathrm{a})=0, \mathrm{f}(\mathrm{b})=1$,
$\mathrm{f}(\mathrm{c})=-1, \mathrm{f}(\mathrm{d})=0$, then the minimum number of zeroes $g(x)=\left(f^{\prime}(x)\right)^{2}+f(x) f^{\prime}(x)$ in $[a, d]$ is $4 .$
Reason (R) : If $f(\alpha) f(\beta)<0$ then $f(\gamma)=0$ for some $\alpha<\gamma<\beta$ and if $f(\alpha)=f(\beta)=0$
then $f^{\prime}(\gamma)=0$ for some $\alpha<\gamma<\beta$

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that for any continuous function defined on a closed interval [?, ?], if the function takes on two values at the endpoints that have opposite signs, then it must take on zero at some point within that interval. This theorem is key in asserting the existence of a zero when a function’s values change sign over an interval and is fundamental in proving the existence of roots in continuous functions.

Rolle's Theorem

Rolle's Theorem is a special case of the Mean Value Theorem, stating that if a function is continuous on a closed interval [?, ?], differentiable on the open interval (?, ?), and takes the same value at the endpoints (i.e. f(?) = f(?)), then there exists at least one c in (?, ?) such that the derivative f'(c) is zero. This theorem provides a method to infer the existence of stationary points between two points where the function values are equal.

Relationship Between Zeroes of a Function and Its Derivative

In the context of differential calculus, there is an important relationship between the zeroes of a differentiable function and its derivative. If a function has a change in sign or returns to the same value over an interval, the IVT and Rolle's Theorem can be used to establish the existence of a zero of the function or its derivative, respectively. This interconnectedness is crucial when analyzing patterns in oscillatory behaviors or when determining the minimum number of zeroes in combinations of functions and their derivatives.

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