Let f(x)={ x^3+x^2+3 x+sin x |(3+sin 1 / x), x ≠ 0 0 , x=0 . then number of po

step 1 Let's solve this question. We have G of X is equal to X cube plus X squared plus 3x plus x plus

Let f(x)={ x^3+x^2+3 x+sin x |(3+sin 1 / x), x ≠ 0 ...

Let $f(x)=\left\{\begin{array}{cl}x^{3}+x^{2}+3 x+\sin x \mid(3+\sin 1 / x), & x \neq 0 \\ 0 & , x=0\end{array}\right.$
then number of points (where $f(x)$ attains its minimum value) is
(A) $]$
(B) $\underline{2}$
(C) 3
(D) infinite many

Key Concepts

Limit Analysis

Limit analysis involves studying the behavior of a function as the input approaches a particular value. This is especially important for points where the function might not be defined in the standard way, as with oscillatory components or piecewise definitions. By evaluating limits, one can determine the tendency of the function near critical points and thus understand its local extrema.

Piecewise?Defined Functions

A piecewise?defined function is one that is described by different formulas over different parts of its domain. When analyzing such functions, special attention must be paid to the boundary points where the definition changes, as these points can be sources of discontinuity or non?differentiability, which in turn affect properties like minima and maxima.

Global Minimum of a Function

The global minimum of a function is the smallest value that the function attains over its entire domain. Finding the global minimum typically involves checking critical points where the derivative is zero or undefined, as well as evaluating the function at the endpoints or points of definition change, ensuring that no other part of the function dips lower.

Continuity at Critical Points

Continuity is a key property that affects the behavior of functions, especially around points where the function’s definition changes. When a function is continuous at a point—particularly at the junction of a piecewise definition—it ensures that the limits from both sides coincide with the function’s value at that point, which is crucial in identifying whether a minimum is actually attained there.

Oscillatory Behavior in Functions

Oscillatory behavior, often introduced by trigonometric functions such as sin(1/x), means that the function exhibits rapid fluctuations as the variable approaches a particular value (like zero). This characteristic can complicate the analysis of limits and continuity, making it necessary to carefully assess the impact of these oscillations on the overall behavior and the attainment of minima or maxima in the function.

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