Nishtha runs a chain of famous restaurants, ‘Indian Kitchen’, in different parts of Delhi. The restaurant chai

Question 19 of 26

Q. Nishtha runs a chain of famous restaurants, ‘Indian Kitchen’, in different parts of Delhi. The restaurant chain was known for good quality food and quick service. ‘Indian Kitchen’ was generating good revenue and had goodwill in the market. Nishtha would take a weekly report from all the Restaurant Managers and take necessary action to resolve the problems, if any. Nishtha noticed a sudden decline in sales in one of the restaurants. She questioned the Restaurant Manager who mentioned that the restaurant is seeing a gradual decrease in the number of customers, but could not provide a clear explanation for it. So, Nishtha formed a team to find out the causes of the declining customers as this is the key area for restaurant business and is critical to the success of business. By doing this, Nishtha is performing a step of the process of a function of management.

(i) Identify and explain the function of management discussed above.

(ii) Explain the step of the function identified in

(i) above, followed by Nishtha to find out the causes of declining revenue.

(iii) Explain the step Nishtha will take after she gets the report from the team formed to find out the causes of declining revenue.

Assam AhsecCBSE Class XII Board 2024Subjective· 6mImportance★★★★★

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Concept understanding — Controlling Definition

Controlling Definition — First Encounter

Imagine you're trying to prove that a function is continuous at a point. You've seen the epsilon-delta definition: for every $\epsilon > 0$ , there exists a $\delta > 0$ such that if $|x - c| < \delta$ , then $|f(x) - f(c)| < \epsilon$ . That's the standard definition.

Now think about what you're actually doing when you use this definition to prove continuity. You're given an arbitrary $\epsilon$ — a tolerance on the output — and you must produce a $\delta$ — a window on the input — that controls the output error. The $\delta$ you choose depends on $\epsilon$ , on the function $f$ , and on the point $c$ . The core idea: you control the input to control the output.

That's the intuition behind the term "controlling definition." It's a definition that tells you: to achieve a desired precision in the output, you must restrict the input within some range. The definition itself gives you the control — the $\delta$ — as a function of the desired precision $\epsilon$ .

The Precise Statement

A controlling definition is a definition of a limit (or continuity, or derivative) that explicitly quantifies the relationship between an output tolerance and an input restriction. For the limit $\lim_{x \to c} f(x) = L$ , the controlling definition is:

\lim_{x \to c} f(x) = L \iff \forall \epsilon > 0, \ \exists \delta > 0 \ \text{such that} \ 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon

The word "controlling" highlights that $\delta$ is chosen after $\epsilon$ is given — you control the input window $\delta$ to force the output error below $\epsilon$ . The $\delta$ is not arbitrary; it must be found (or shown to exist) for every possible $\epsilon$ .

Why "Controlling" Matters

This phrasing is most common in analysis textbooks when contrasting with other equivalent definitions. For example, the "sequential definition" of a limit says: if every sequence $x_n \to c$ (with $x_n \neq c$ ) has $f(x_n) \to L$ , then the limit exists. That definition doesn't give you a direct control mechanism — it's about checking sequences.

The controlling definition is the one you actually use in proofs. It's constructive: you pick $\delta$ as a function of $\epsilon$ , often by solving inequalities. For instance, to prove $\lim_{x \to 2} (3x+1) = 7$ , you set $\delta = \epsilon/3$ . That's the control: if $|x-2| < \epsilon/3$ , then $|(3x+1)-7| = 3|x-2| < \epsilon$ . …

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Nishtha is performing the Controlling function of management, specifically the step of Analyzing Deviations by forming a team to find the causes of declining sales, and her next step will be Taking Corrective Action based on their report.

The scenario describes Nishtha's proactive approach to managing her restaurant chain, which is a classic example of a fundamental management function. Management involves a series of interconnected functions: Planning, Organising, Staffing, Directing, and Controlling. These functions work in a cycle, ensuring that an organisation moves towards its objectives efficiently and effectively.

In this case, Nishtha's actions are geared towards ensuring that the actual performance (sales) aligns with the planned or expected performance. When she notices a deviation (decline in sales), her subsequent actions to investigate and resolve the issue fall squarely within the scope of the 'Controlling' function. The intuition behind controlling is simple: you set a goal, you monitor progress towards that goal, and if you're off track, you figure out why and fix it. Without control, even the best plans can fail because there's no mechanism to ensure execution.

Let's break down Nishtha's actions according to the steps of the controlling process.

Identify and explain the function of management discussed above.

The function of management discussed is Controlling.

Controlling is the process of ensuring that actual activities conform to planned activities. It involves setting performance standards, measuring actual performance, comparing actual performance with standards, identifying deviations, and taking corrective actions. The primary purpose of controlling is to ensure that organisational objectives are achieved by keeping the organisation on the right track. Nishtha's weekly reports are a form of performance measurement, and her reaction to the decline in sales (a deviation from expected performance) by investigating its causes and planning further action is the essence of the controlling function.
Explain the step of the function identified in (i) above, followed by Nishtha to find out the causes of declining revenue.

The specific step of the controlling function that Nishtha is performing is Analyzing Deviations.

After comparing actual performance (declining sales) with the standard (expected or previous sales levels) and identifying a significant deviation, the next crucial step is to analyze these deviations to determine their causes. It's not enough to just know that sales are down; one must understand why. Nishtha first questioned the Restaurant Manager, who could not provide a clear explanation. This indicated that a deeper investigation was needed. By forming a team to find out the causes, Nishtha is actively engaging in analyzing the deviations. This step involves a thorough investigation to pinpoint the root causes of the problem, which could range from changes in customer preferences, quality issues, service problems, increased competition, or even external factors. …

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