Let S be the set of points where the function, f (x) = |2 - |x-3||, x ∈ R is not differentiable. The

Q: Wait, why do we only look for kinks? Can't a function be non-differentiable elsewhere?

That is a brilliant question! For a continuous function like this one, the only points where the derivative fails to exist are indeed the sharp corners, or 'kinks.' Since our function is a composition of continuous modulus functions, it is continuous everywhere. Therefore, the kinks are the only culprits. If the function were discontinuous, that would be a different story, but here, we only need to hunt for those sharp turns.

Q: I found $x=3$ and $x=5$, but I missed $x=1$. Why is that?

Ah, the classic trap! When you solve the equation $|2 - |x-3|| = 0$, you get $|x-3| = 2$. This is an absolute value equation of the form $|A| = k$, which implies $A = k$ or $A = -k$. So, you must solve $x-3 = 2$ AND $x-3 = -2$. Both are valid and yield distinct points. Always remember to split the modulus equation into both positive and negative cases!

Q: Do I really have to calculate $f(f(x))$ for every point? Is there a shortcut?

While it is tempting to look for a pattern, with only three points, direct calculation is the safest and most reliable path. It prevents silly errors and builds confidence. Trust the process—calculate $f(1)$, then $f(f(1))$, and so on. It is a small price to pay for absolute certainty in your answer.

Question

Let $S$ be the set of points where the function, $f (x) = |2 - |x-3||$, $x \in R$ is not differentiable. Then $\sum_{x \in S} f (f(x))$ is equal to. . . . .

Sigma Percentile · Accepted Answer

### The Geometry of Sharp Turns: A Journey into Non-Differentiability

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving an equation; we are dissecting the anatomy of a function.

We are looking at $f(x) = |2 - |x-3||$. This is a beautiful, nested structure that folds onto itself, creating a landscape of sharp peaks and valleys.

Our mission is to find the set $S$ of points where this function refuses to be smooth—where the derivative simply does not exist—and then perform a final summation. Let us begin.

### The Anatomy of a Kink

Imagine standing on the graph of $y = |x|$. As you walk along the negative $x$-axis, you are moving with a slope of $-1$.

Suddenly, at $x=0$, you hit a sharp corner. You cannot define a single tangent line there because the slope jumps instantly from $-1$ to $+1$.

This is the essence of non-differentiability. In our function $f(x) = |2 - |x-3||$, we have two layers of these 'V' shapes.

Algebraically, non-differentiability occurs whenever the expression inside a modulus evaluates to zero. It is the moment the function 'decides' to change direction.

### The Hunt for the Set $S$

Let us start from the inside out. The innermost expression is $x-3$. Setting this to zero gives us $x-3 = 0$, which implies $x = 3$. This is our first point of non-differentiability.

Now, we move to the outer layer. We set the entire expression inside the outer modulus to zero:
$$2 - |x-3| = 0$$

This rearranges to $|x-3| = 2$. Remember, an absolute value equation $|A| = k$ splits into two possibilities: $A = k$ or $A = -k$.

So, we have two cases: $x-3 = 2$ and $x-3 = -2$. Solving these, we get $x = 5$ and $x = 1$.

Thus, our set $S$ of non-differentiable points is $S = \{1, 3, 5\}$. We have successfully mapped the 'kinks' of our function.

### The Nested Evaluation

Now, we must evaluate the nested function $f(f(x))$ for each point in $S$. Let us take them one by one.

For $x = 1$, we first find:
$$f(1) = |2 - |1-3|| = |2 - |-2|| = |2 - 2| = 0$$

Now, we need $f(f(1))$, which is $f(0)$. Substituting $x=0$ into our original function:
$$f(0) = |2 - |0-3|| = |2 - 3| = |-1| = 1$$

So, $f(f(1)) = 1$. By symmetry, $x = 5$ will yield the same result:
$$f(5) = |2 - |5-3|| = |2 - 2| = 0 \Rightarrow f(f(5)) = f(0) = 1$$

Finally, we tackle the peak at $x = 3$. We calculate:
$$f(3) = |2 - |3-3|| = |2 - 0| = 2$$

Now, we need $f(f(3))$, which is $f(2)$. Substituting $x=2$ into our function:
$$f(2) = |2 - |2-3|| = |2 - |-1|| = |2 - 1| = 1$$

So, $f(f(3)) = 1$.

### The Final Summation

We have arrived at the finish line. We have found that for every point in our set $S$, the value of the nested function $f(f(x))$ is $1$.

The summation is:
$$\sum_{x \in S} f(f(x)) = f(f(1)) + f(f(3)) + f(f(5)) = 1 + 1 + 1 = 3$$

The elegance of the result is satisfying, isn't it? The final answer is **3**.

Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $S$ be the set of points where the function, $f (x) = ∣2 - ∣ x - 3∣∣$ , $x \in R$ is not differentiable. Then $\sum_{x \in S} f (f (x))$ is equal to. . . . .

Enter Numerical Value:

Visualized Solution

Visualizing the Function $f (x)$

Condition for Non-Differentiability

Analyzing the Inner Modulus

Analyzing the Outer Modulus

Solving for $x$ in the Outer Modulus

Identifying the Set $S$

Evaluating $f (1)$ and $f (5)$

Evaluating $f (3)$

Calculating $f (f (1))$ and $f (f (5))$

Calculating $f (f (3))$

Final Summation

The Sigma Insight: Differentiability of a Function

The Geometry of Sharp Turns

The Anatomy of a Kink

The Hunt for the Set $S$

The Nested Evaluation

The Final Summation

Similar Questions

Let $S$ be the set of points where the function, $f (x) = ∣2 - ∣ x - 3∣∣, x \in R$ , is not differentiable. Then, the value of $\sum_{x \in S} f (f (x))$ is equal to

The number of points, where the function $f : R \to R$ , $f (x) = ∣ x - 1∣ cos ∣ x - 2∣ sin ∣ x - 1∣ + (x - 3) ∣ x^{2} - 5 x + 4∣$ , is NOT differentiable, is :

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Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

Let $f : (- 1, 1) \to R$ be a function defined by $f (x) = max {- ∣ x ∣, - 1 - x^{2}}$ . If K be the set of all points at which f is not differentiable, then K has exactly :

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{2}}$ . Let $S$ denote the set of all points in $R$ , where $f$ is not differentiable. Then :

Let $K$ be the set of all real values of $x$ where the function $f (x) = sin ∣ x ∣ - ∣ x ∣ + 2 (x - π) cos ∣ x ∣$ is not differentiable. Then the set $K$ is equal to:

Let $f (x) = 15 - ∣ x - 10∣; x \in R$ . Then the set of all values of $x$ , at which the function, $g (x) = f (f (x))$ is not differentiable, is :

Let $S = {t \in R : f (x) = ∣ x - π ∣ \cdot (e^{∣ x ∣} - 1) sin ∣ x ∣ is not differentiable at t}$ . Then the set $S$ is equal to :

Enter Numerical Value:

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)

Condition for Non-Differentiability

Analyzing the Inner Modulus

Analyzing the Outer Modulus

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x in the Outer Modulus

Identifying the Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(1) and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(5)

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(3)

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(f(1)) and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(f(5))

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(f(3))

Final Summation

The Sigma Insight: Differentiability of a Function

The Geometry of Sharp Turns

The Anatomy of a Kink

The Hunt for the Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

The Nested Evaluation

The Final Summation

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a function defined by f(x)=max{x,x3}. The set of all points where f(x) is NOT differentiable is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:(−1,1)→R be a function defined by f(x)=max{−∣x∣,−1−x2​}. If K be the set of all points at which f is not differentiable, then K has exactly :

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