The image contains statements about the differentiability of a function $f(x)$ at a point $x = a$ and the relationship between $f(x)$, $g(x)$, and $h(x)$. Let's analyze the statements step by step:

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1. Statement 1:

$f(x) \text{ is differentiable at } x = a \implies |f(x)| \text{ is differentiable at } x = a$

Analysis:

This statement is incorrect. The absolute value function $|f(x)|$ may not be differentiable at $x = a$, even if $f(x)$ is differentiable. For example, if $f(a) = 0$, the derivative of $|f(x)|$ at $x = a$ depends on the behavior of $f(x)$ near $a$, which can cause a cusp in $|f(x)|$.

Conclusion:

The answer is not true.

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2. Statement 2:

$f(x) = g(x) + h(x), \text{ and } g(x) \text{ or } h(x) \text{ is not differentiable at } x = a \implies f(x) \text{ is not differentiable at } x = a$

Analysis:

This statement is correct. Differentiability of $f(x)$ at $x = a$ requires both $g(x)$ and $h(x)$ to be differentiable at $x = a$. If either $g(x)$ or $h(x)$ is not differentiable, their sum $f(x)$ will not be differentiable.

Conclusion:

The answer is true.

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3. Statement 3:

$f(x) \text{ is differentiable at } x = a \implies |f(x)| \text{ is differentiable at } x = a$

Analysis:

This repeats the logic of Statement 1 and is also incorrect. As explained earlier, $|f(x)|$ is not necessarily differentiable at $x = a$, even if $f(x)$ is differentiable, particularly when $f(a) = 0$.

Conclusion:

The answer is not true.

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Final Results:

1. Statement 1: Not True
2. Statement 2: True
3. Statement 3: Not True

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Would you like a detailed explanation for any part of the analysis, or examples to clarify?

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Here are 5 related questions for further exploration:

1. What are the conditions for differentiability of a composite function?
2. How does the chain rule apply to absolute value functions?
3. Can a piecewise function be differentiable everywhere?
4. How is continuity related to differentiability?
5. What happens when a function's derivative approaches infinity?

Tip: Always check the derivative of each component function separately to ensure overall differentiability.