Consider f, g and h be three real valued dilierentiable...

Consider $f, g$ and $h$ be three real valued dilierentiable functions defined on $\mathrm{R}$. Let $g(x)=x^{3}+g^{\prime \prime}(1) x^{2}+\left(3 g^{\prime}(1)-g^{\prime \prime}(1)-1\right) x+3 g^{\prime}(1)$, $f(x)=x g(x)-12 x+1$ and $f(x)=(h(x))^{2}$ where $h(0)=1$. Which of the following is are true for the function $\mathrm{y}=\mathrm{g}(\mathrm{x})$ ? (A) $g(x)$ monotonically decreases in $\left(-\infty, 2-\frac{1}{\sqrt{3}}\right)$ and $\left(2+\frac{1}{\sqrt{3}}, \infty\right)$ (B) g(x) monotonically increases in $\left(2-\frac{1}{\sqrt{3}}, 2+\frac{1}{\sqrt{3}}\right)$ (C) There exists exactly one tangent to $\mathrm{y}=\mathrm{g}(\mathrm{x})$ which is parallel to the chord joining the points $(1, g(1))$ and $(3, g(3))$ (D) There exists exactly two distinct Lagrange's Mean Value in $(0,4)$ for the function $\mathrm{y}=\mathrm{g}(\mathrm{x})$.

Consider $f, g$ and $h$ be three real valued dilierentiable functions defined on $\mathrm{R}$.
Let $g(x)=x^{3}+g^{\prime \prime}(1) x^{2}+\left(3 g^{\prime}(1)-g^{\prime \prime}(1)-1\right) x+3 g^{\prime}(1)$, $f(x)=x g(x)-12 x+1$ and $f(x)=(h(x))^{2}$ where $h(0)=1$.
Which of the following is are true for the function $\mathrm{y}=\mathrm{g}(\mathrm{x})$ ?
(A) $g(x)$ monotonically decreases in $\left(-\infty, 2-\frac{1}{\sqrt{3}}\right)$ and $\left(2+\frac{1}{\sqrt{3}}, \infty\right)$
(B) g(x) monotonically increases in $\left(2-\frac{1}{\sqrt{3}}, 2+\frac{1}{\sqrt{3}}\right)$
(C) There exists exactly one tangent to $\mathrm{y}=\mathrm{g}(\mathrm{x})$ which is parallel to the chord joining the points $(1, g(1))$ and $(3, g(3))$
(D) There exists exactly two distinct Lagrange's Mean Value in $(0,4)$ for the function $\mathrm{y}=\mathrm{g}(\mathrm{x})$.

Key Concepts

Differentiability and Derivatives

Differentiability ensures that a function has a well?defined derivative at each point, which is essential in analyzing its behavior. The derivative, being the instantaneous rate of change, is used to determine where the function is increasing or decreasing—information that is vital for understanding monotonicity and for applying other theorems such as the Mean Value Theorem.

Monotonicity Analysis

Monotonicity of a function refers to its behavior of being entirely non?increasing or non?decreasing over an interval. By examining the sign of the derivative, one can establish intervals where the function is monotonically increasing (derivative positive) or decreasing (derivative negative). This concept is key when analyzing the behavior of polynomial functions and understanding the nature of their turning points.

Mean Value Theorem

The Mean Value Theorem is a fundamental result in calculus that guarantees, given a function continuous on a closed interval and differentiable on the interior, the existence of at least one point where the tangent slope (derivative) equals the average slope between the endpoints. In problems involving tangents parallel to a chord or multiple instances of such occurrences, the MVT serves as a theoretical basis for determining how many points satisfy these conditions within a given interval.

Tangent Lines and Parallelism

Studying tangent lines and their slopes is crucial for understanding linear approximations and local behavior of functions. When a tangent line is parallel to a chord joining two points on a function, its slope is equal to the difference quotient of those two points. This concept is used to identify unique or multiple points where the derivative matches a specific value, often in the context of verifying results from the Mean Value Theorem.

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