
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

A random variable \( X \) is defined by the distribution function

$$
F(x)=\left\{\begin{array}{lll}
0 & \text{for} & x \leq 2 \\
(x-2)^{2} & \text{for} & 2 < x \leq 3 \\
1 & \text{for} & x > 3
\end{array}\right.
$$

Find:

a) The probability density function \( f(x) \);

b) The graphs of the functions \( F(x) \) and \( f(x) \);

c) Using the known function \( F(x) \) and the found function \( f(x) \), the probability that the outcome \( X \) will be at least 2.1 and less than 2.5.

Provide a geometric interpretation of the probability \( P(2.1 \leq x < 2.5) \).

Present the answer in LaTex format: \boxed{Your answer}
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