Column-I Column-II I. If f(x)= (A) R-[-√(3), √(3)] { 4 x-x^3+ln(a^2-3 a+3), 0 ≤ x<3 x-

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Column-I Column-II I. If f(x)= (A) R-[-√(3), √(3)] { 4...

Column-I Column-II I. If $f(x)=$ (A) $R-[-\sqrt{3}, \sqrt{3}]$ $\left\{\begin{array}{c}4 x-x^{3}+\ln \left(a^{2}-3 a+3\right), 0 \leq x<3 \\ x-18, & x \geq 3\end{array}\right.$ $=$ has a local minima at $x=3$, than $a$ belongs to II. If the function $f(x)=$ (B) $(0, \infty)$ $\left(\frac{\sqrt{a+1}}{a-1}-1\right) x^{3}-x+\ln (a-1)$ is strictly decreasing $\forall x \in R$, then $a$ belongs toIII. If the function $f(x)=x^{3}+$ (C) $[1,2]$ $a x^{2}+a^{2} x+2 \sin ^{2} x$ is strictly increasing $\forall x \in R$, then $a$ belongs to IV. The function $f(x)=\left|e^{a x}-e^{-a x}\right|$, (D) $(3, \infty)$ $a>0$ is strictly increasing in the interval

Column-I Column-II
I. If $f(x)=$
(A) $R-[-\sqrt{3}, \sqrt{3}]$
$\left\{\begin{array}{c}4 x-x^{3}+\ln \left(a^{2}-3 a+3\right), 0 \leq x<3 \\ x-18, & x \geq 3\end{array}\right.$
$=$ has a local minima at $x=3$, than
$a$ belongs to
II. If the function $f(x)=$
(B) $(0, \infty)$
$\left(\frac{\sqrt{a+1}}{a-1}-1\right) x^{3}-x+\ln (a-1)$
is strictly decreasing $\forall x \in R$, then $a$ belongs toIII. If the function $f(x)=x^{3}+$
(C) $[1,2]$ $a x^{2}+a^{2} x+2 \sin ^{2} x$ is strictly
increasing $\forall x \in R$, then $a$ belongs to
IV. The function $f(x)=\left|e^{a x}-e^{-a x}\right|$,
(D) $(3, \infty)$
$a>0$ is strictly increasing in the interval

Key Concepts

Parameter Influence on Function Behavior

Many functions include parameters whose values affect characteristics like local extrema, monotonicity, and overall behavior. Analyzing how these parameters influence the function often involves setting up inequalities or derivative conditions and solving them under the constraints provided (such as domain restrictions) to determine the range of acceptable parameter values. This concept is central in understanding how modifications in the function's formulation impact its overall characteristics.

Piecewise Function Analysis

Piecewise functions are defined by different expressions over different intervals, and analyzing their behavior, especially at the boundaries where the expression changes, requires careful consideration of continuity, differentiability, and the matching of derivative values where applicable. Conditions such as local minima at the boundary points often involve ensuring that the limits from each side and the derivative behavior are consistent with the desired property.

Monotonicity

Monotonicity describes functions that are exclusively non-decreasing or non-increasing over an interval. A function is said to be strictly increasing if for every pair of points the later function value is greater than the earlier one, and strictly decreasing if it is lesser. This property is typically established by analyzing the sign of the derivative over the entire domain, and it is key when determining the nature of functions with respect to order and transformation.

Derivative Test

The derivative test is a fundamental tool in calculus used to determine whether a function has a local minimum, local maximum, or neither at a given point. By computing the first derivative of a function and analyzing its sign or applying the second derivative test, one can assess the behavior of the function around critical points. This test is critical when evaluating both extremal points and overall monotonicity.

Local Extrema

Local extrema refer to the minimum or maximum values that a function takes in a small neighborhood around a point. Determining whether a point is a local minimum or maximum often involves checking conditions such as the first derivative being zero (or not existing) at that point and verifying the behavior of the function around that point (for instance, using the second derivative test or analyzing the sign change of the first derivative). This concept is essential in optimization and in the study of function behavior.

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