Question

If $\underset{h\to 0}{lim}\left[\frac{f(a+h)}{f(a)}\right]=c$ where $f(x)$ is a continuous function such that $f(x)>0$ for all $x\in R$ and $[.]$ denotes greatest integer function then which of the following statements is always true? (1) If $x=a$ is a point of local minima then $c\in N$ (2) If $x=a$ is a point of local maxima then $c\in N$ (3) If $x=a$ is a point of local minima then $c\in I^{-}$ (4) If $x=a$ is a point of local maxima then $c\in I^{-}$

Short answer

Option (2) is always true: If $x=a$ is a point of local maxima then $c\in N$.

Step-by-step solution

- Understand the Given Limit

The problem states that the limit $${\text{lim}}_{h\to 0}\left[\frac{f(a+h)}{f(a)}\right]=c$$ Here, $$[\cdot ]$$ is the greatest integer function which means it rounds down the expression to the nearest integer.

- Analyzing Behavior of Continuous Function

Since $f(x)$ is continuous and positive for all $x\in \mathbb{R}$, the function is guaranteed to change smoothly without breaks or jumps as $h$ approaches 0.

- Relate Limit to Continuity at Local Extrema

For $x=a$ being a point of local minima or maxima, the limit indicates the behavior around $a$. Let us consider two cases: **Local Minima:** At a local minimum, $f(a)$ is less than $f(a+h)$, thus $\frac{f(a+h)}{f(a)}\ge 1$. **Local Maxima:** At a local maximum, $f(a)$ is greater than $f(a+h)$, thus $\frac{f(a+h)}{f(a)}\le 1$.

- Determine the Value of c with Floor Function

Greatest integer function $[\cdot ]$ implies that $c$ must be the greatest integer below or equal to the ratio.For local minima: $\left[\frac{f(a+h)}{f(a)}\right]\ge 1$. If the limit is always an integer, $c$ must be a non-negative integer. For local maxima: positive function stays $\left[\frac{f(a+h)}{f(a)}\right]\le 1$ indicating $c\le 1$, with it typically below 1 due to floor rounding down and often making $c\le 0$.

- Choose the Correct Option

Comparing the given statements:Assessing both local minimum and maximum scenarios primarily gives us insights on nature of $c$ considering valid limits and behaviors. Since positive behavior typically sustains, thus correct interpretation should lean accurate indicating only integer fits is close to actual behavior.Hence, statement (2) ensures correctness in standardized outcome making $x=a$ maxima ensuring integral exact matching.

Key concepts

Local Extrema Behavior

In calculus, local extrema (local minima and maxima) play a significant role in understanding the behavior of functions. A local minimum occurs at a point where the function value is lower than at any other nearby point. Conversely, a local maximum occurs where the function value is higher than at any nearby point.

At a **local minimum** around point 'a', the function value at 'a' ($f(a)$) is less than the values of the function in its immediate vicinity ($f(a+h)$) as we move a small distance 'h' from 'a'. In mathematical terms, this ensures that the function ratio $\frac{f(a+h)}{f(a)}$ must be greater than or equal to 1. This suggests that the corresponding integer value would also be an integer greater or equal to 1 due to the greatest integer function.

For a **local maximum**, a similar concept applies: the function value at the point 'a' ($f(a)$) is higher than those nearby ($f(a+h)$). Consequently, the ratio $\frac{f(a+h)}{f(a)}$ must be less than or equal to 1. This means that the greatest integer function applied here would return an integer less than or equal to 1.

This behavior allows us to infer that the behavior of 'c' (as explained in the question) aligns closely with integral considerations as we analyze local characteristics around these extrema points.

Greatest Integer Function

The greatest integer function, often denoted as $$\text{floor}(x)$$ or $$[x]$$, rounds down any real number 'x' to the nearest lowest integer.

For instance, $$[3.7]=3$$ and $$[-1.2]=-2$$. When applied within calculations, this function influences results by ensuring always to consider the next lowest integer effectively. This has vital significance in certain limit problems where approximations converging to integer values are desired.

The importance of the greatest integer function becomes evident especially when we’re dealing with continuous functions around specific points such as local minima and maxima. For instance, as we approach 'h' closer to zero, $\frac{f(a+h)}{f(a)}$ would usually converge towards 1, and the floor function would adjust it to fit within pre-established boundary constraints as derived in integer form.

This linking mechanism between evaluated ratios and integral outcomes helps maintain intact the rational expectations and properties embodied behind those observations aptly adjusting real-time characteristics into structured integral forms.

Limits in Calculus

Limits are fundamental in calculus, providing the basis to evaluate how functions behave as inputs approach specific values. A limit ${\text{lim}}_{h\to 0}f(x+h)$ implies determining the function's value as 'h' gets infinitesimally small, i.e., approaching a specific point.

The given exercise problem involves this concept telling us to consider the value of the limit: ${\text{lim}}_{h\to 0}\frac{f(a+h)}{f(a)}=c$. Here 'c' is derived based on the limit proportion and continuity rule frameworks.

Continuity assurance implies that within an infinitely small vicinity around 'a', the function progresses smoothly assuring well-defined values around its shifting corridor ensuring bounded limits typically integral due to innate stability.

This interplay of ratios, floor-adjusted outcomes, and apparent continuity defines robust and consistent derivative behaviors solidifying structured perspectives on extremal behaviors such as maxima, and minima accurately reflecting captured integral output nuances.