Consider the function, f(x) = |x - 2| + |x - 5|, x in R. Statement-1 : f'(4) = 0 Statement-2 : f is

Q: Wait, why is the derivative zero? I thought absolute value functions always have sharp corners where the derivative is undefined?

That is a brilliant observation! You are absolutely right that at the 'hinge' points $x = 2$ and $x = 5$, the function is not differentiable because of those sharp corners. However, the question asks about $f'(4)$. Since $x = 4$ is strictly inside the interval $(2, 5)$, we are far away from those sharp corners. In that specific region, the function behaves like a flat, horizontal line, which is why the derivative is zero.

Q: Is Statement-2 really the *explanation* for Statement-1? It feels like a coincidence.

It is not a coincidence at all! Statement-2 provides the formal, theoretical framework—Rolle's Theorem—that guarantees the existence of a point where the derivative is zero. Because our function satisfies all the conditions of Rolle's Theorem on the interval $[2, 5]$, it mathematically necessitates that there must be at least one point $c$ in $(2, 5)$ where $f'(c) = 0$. Since $4$ is one such point, Rolle's Theorem is indeed the underlying reason.

Q: Do I need to check the derivative at the endpoints $x = 2$ and $x = 5$?

Great question. For the purpose of evaluating Statement-1, we only care about $x = 4$, which is in the open interval $(2, 5)$. For Statement-2, we only need to check the conditions of Rolle's Theorem, which specifically requires continuity on the closed interval $[2, 5]$ and differentiability on the open interval $(2, 5)$. We do not need to worry about the derivative at the endpoints themselves.

Question

Consider the function, $f(x) = |x - 2| + |x - 5|, x \in R$. Statement-1 : $f'(4) = 0$. Statement-2 : $f$ is continuous in $[2, 5]$, differentiable in $(2, 5)$ and $f(2) = f(5)$.

Options:

(A) Statement-1 is false, Statement-2 is true.

(B) Statement-1 is true, Statement-2 is true; statement-2 is a correct explanation for Statement-1.

(C) Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1.

(D) Statement-1 is true, Statement-2 is false.

Sigma Percentile · Accepted Answer

### Analyzing the Landscape

Imagine you are standing on a vast, flat plain, and suddenly, you encounter a V-shaped valley. This is the geometric reality of the function $f(x) = |x - 2| + |x - 5|$.

The points $x = 2$ and $x = 5$ are the 'hinges' of this landscape, the places where the ground shifts beneath your feet. To understand the behavior of this function, we must walk through the interval between these two hinges: the interval $[2, 5]$.

### The Magic of the Flat Zone

When we step into the interval $[2, 5]$, something fascinating happens. For any $x$ such that $2 \le x \le 5$, the term $(x - 2)$ is non-negative, so $|x - 2|$ simply becomes $(x - 2)$.

Simultaneously, the term $(x - 5)$ is non-positive, so $|x - 5|$ becomes $-(x - 5)$. When we add these together, we get:

$$f(x) = (x - 2) - (x - 5)$$

Watch closely as the algebra unfolds:

$$f(x) = x - 2 - x + 5 = 3$$

The variable $x$ vanishes! We are left with $f(x) = 3$. This is the 'Flat Zone'. Between $x = 2$ and $x = 5$, the function is not climbing or falling; it is perfectly horizontal at a height of $3$.

This explains why $f'(4) = 0$. Since the function is constant in this region, its rate of change is zero. **Statement-1 is undeniably true.**

### The Theoretical Bridge: Rolle's Theorem

Now, let us turn our attention to Statement-2. It asks us to verify three conditions: continuity on $[2, 5]$, differentiability on $(2, 5)$, and the equality $f(2) = f(5)$.

First, continuity: absolute value functions are the building blocks of continuous curves, and their sum is also continuous.

Second, differentiability: we have already seen that in the open interval $(2, 5)$, the function is a constant, and constants are the smoothest functions of all—they are infinitely differentiable.

Third, the endpoints:

$$f(2) = |2 - 2| + |2 - 5| = 0 + 3 = 3$$

$$f(5) = |5 - 2| + |5 - 5| = 3 + 0 = 3$$

Thus, $f(2) = f(5)$. These three conditions are the exact requirements for **Rolle's Theorem**.

### Conclusion

Rolle's Theorem is a powerful guarantee in calculus; it tells us that if a function is continuous, differentiable, and starts and ends at the same height, there must be a point in between where the slope is zero.

By stating these conditions, Statement-2 provides the theoretical foundation that explains why $f'(4) = 0$. It is not just a set of observations; it is the logical proof for the behavior we observed.

Therefore, **Statement-2 is not only true but is the correct explanation for Statement-1.**

Animated Solution for Mathematics - Differentiation: Consider the function, $f (x) = ∣ x - 2∣ + ∣ x - 5∣, x \in R$ . Statement-1 : $f^{'} (4) = 0$ . Statement-2 : $f$ is continuous in $[2, 5]$ , differentiable in $(2, 5)$ and $f (2) = f (5)$ .

Select Answer:

Visualized Solution

Visualizing the Function $f (x)$

Defining the Interval $[2, 5]$

Resolving Absolute Values

Raw Setup for $f (x)$

Simplifying $f (x)$

Graphing the Interval

Evaluating Statement-1

Checking Statement-2: Continuity

Checking Statement-2: Differentiability

Checking Statement-2: Endpoints

Connecting to Rolle's Theorem

Conclusion

The Sigma Insight: Mean Value Theorems

Analyzing the Landscape

The Magic of the Flat Zone

The Theoretical Bridge

Conclusion

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Select Answer:

Visualized Solution

Visualizing the Function .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Defining the Interval .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } [2,5]

Resolving Absolute Values

Raw Setup for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Simplifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Graphing the Interval

Evaluating Statement-1

Checking Statement-2: Continuity

Checking Statement-2: Differentiability

Checking Statement-2: Endpoints

Connecting to Rolle's Theorem

Conclusion

The Sigma Insight: Mean Value Theorems

Analyzing the Landscape

The Magic of the Flat Zone

The Theoretical Bridge

Conclusion

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Raw Setup for $f (x)$

Simplifying $f (x)$

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