If |f(x1) - f(x2)|

Q: Wait, why does the strict inequality $<$ turn into $\le$ when we take the limit? Isn't that mathematically illegal?

It is a fundamental property of limits! If $g(x) < h(x)$ for all $x$ in a neighborhood, then $\lim_{x \to a} g(x) \le \lim_{x \to a} h(x)$. The strict inequality can 'collapse' into equality at the limit point. It is not illegal; it is the definition of how inequalities behave under limits.

Q: Why can't the function be something like $f(x) = x^2$? If I plug that in, does it work?

Great question! Let's test it. If $f(x) = x^2$, then $|f(x_1) - f(x_2)| = |x_1^2 - x_2^2| = |x_1 - x_2| \cdot |x_1 + x_2|$. The inequality requires $|x_1 + x_2| < |x_1 - x_2|$, which is definitely not true for all $x_1, x_2$. It fails! This shows that only a very specific type of function can satisfy this condition.

Q: I understand $f'(x) = 0$, but why does that mean the function is constant? Can't it be something else?

Think about the Mean Value Theorem. If $f'(x) = 0$ on an interval, the slope is zero everywhere. If the slope is zero, the function cannot rise or fall. It must be a horizontal line, which is a constant function. There is no other possibility!

Question

If $|f(x_1) - f(x_2)| < (x_1 - x_2)^2$, for all $x_1, x_2 \in \mathbb{R}$. Find the equation of tangent to the curve $y = f(x)$ at the point $(1, 2)$.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a graph, looking at a curve $y = f(x)$. You are told that the difference between the heights of any two points is always smaller than the square of the horizontal distance between them.

This is expressed by the inequality:
$$|f(x_1) - f(x_2)| < (x_1 - x_2)^2$$

At first glance, this looks like a simple algebraic inequality, but for a JEE student, this is a siren song of calculus. It is a hidden message about the rate of change.

### The Difference Quotient Insight

To understand the behavior of this function, we need to examine its derivative. The derivative is defined as the limit of the difference quotient:
$$\frac{f(x_1) - f(x_2)}{x_1 - x_2}$$

Our given inequality involves the square of the difference, $(x_1 - x_2)^2$. If we divide both sides of our inequality by $|x_1 - x_2|$, assuming $x_1 
eq x_2$, we obtain:
$$\left| \frac{f(x_1) - f(x_2)}{x_1 - x_2} ight| < |x_1 - x_2|$$

Suddenly, the left side looks exactly like the definition of the derivative. We have successfully transformed an abstract inequality into a statement about the slope of the curve.

### The Limit Transition

Now, we perform the magic of calculus. We let $x_1$ approach $x_2$. As the two points get infinitely close, the left side of our inequality becomes the absolute value of the derivative, $|f'(x_2)|$.

On the right side, as $x_1 	o x_2$, the term $|x_1 - x_2|$ vanishes and becomes $0$. Here is the critical transition: the strict inequality $<$ becomes $\le$ in the limit.

So, we arrive at:
$$|f'(x_2)| \le 0$$

Since the absolute value of any real number is always non-negative, the only way for $|f'(x_2)|$ to be less than or equal to zero is if it is exactly zero. Thus, $f'(x) = 0$ for all $x$.

### The Constant Verdict

If the derivative of a function is zero everywhere, it tells us that the function has no slope. It is not climbing and it is not falling; it is perfectly, serenely flat.

A function with a zero derivative everywhere must be a constant function, $f(x) = c$. We have stripped away the mystery; the curve is not a complex parabola or a wild trigonometric function, but a horizontal line.

### The Final Reveal

We are given that the curve passes through the point $(1, 2)$. Since our function is $f(x) = c$, this implies $f(1) = 2$, so $c = 2$.

The function is simply $f(x) = 2$. The tangent to a horizontal line at any point is the line itself.

Therefore, the equation of the tangent is **$y = 2$**. Keep this logic in your toolkit; whenever you see $|f(x_1) - f(x_2)| \le K|x_1 - x_2|^n$ with $n > 1$, you are looking at a constant function.

Animated Solution for Mathematics - Differentiation: If $∣ f (x_{1}) - f (x_{2}) ∣ < (x_{1} - x_{2})^{2}$ , for all $x_{1}, x_{2} \in R$ . Find the equation of tangent to the curve $y = f (x)$ at the point $(1, 2)$ .

Visualized Solution

Visualizing the Point $(1, 2)$

Analyzing the Given Inequality

Rearranging for the Derivative Form

Applying the Limit Definition

Interpreting the Derivative

Identifying the Constant Function

Solving for the Constant

Finding the Equation of the Tangent

Summary and Key Takeaway

The Sigma Insight: Tangents, Normals and Rate Measure

Analyzing the Setup

The Difference Quotient Insight

The Limit Transition

The Constant Verdict

The Final Reveal

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