To solve for $k$ and evaluate the given probabilities, we first need to find the normalization constant to ensure that $f(x)$ is a valid probability density function (PDF).

Given the PDF: $f(x) = k(1 + x) \quad \text{for} \quad 2 < x < 5$

Step 1: Find $k$ such that $f(x)$ is normalized.

The total integral over the interval $2 < x < 5$ should equal 1: $\int_{2}^{5} k(1 + x) \, dx = 1$

Calculating the integral: $k \int_{2}^{5} (1 + x) \, dx = k \left[ x + \frac{x^2}{2} \right]_{2}^{5}$

Evaluating at the bounds: [ k \left[ \left(5 + \frac{5^2}{2}\right) - \left(2 + \frac{2^2}{2}\right) \right] = k \left[ \left(5 + \frac{25}{2}\right) - \left(2 + 2\right) \right] ]

Simplifying further: $k \left[ 5 + 12.5 - 4 \right] = k \cdot 13.5$

Setting this equal to 1: $k \cdot 13.5 = 1 \quad \Rightarrow \quad k = \frac{1}{13.5}$

Step 2: Find $P(x < 4)$

$P(x < 4) = \int_{2}^{4} \frac{1}{13.5}(1 + x) \, dx$

Solving the integral: $\frac{1}{13.5} \int_{2}^{4} (1 + x) \, dx = \frac{1}{13.5} \left[ x + \frac{x^2}{2} \right]_{2}^{4}$

Evaluating at the bounds: [ \frac{1}{13.5} \left[ (4 + \frac{16}{2}) - (2 + 2) \right] = \frac{1}{13.5} \left[ 4 + 8 - 4 \right] = \frac{8}{13.5} ]

Simplifying: $P(x < 4) = \frac{8}{13.5} \approx 0.593$

Step 3: Find $P(3 < x < 4)$

$P(3 < x < 4) = \int_{3}^{4} \frac{1}{13.5}(1 + x) \, dx$

Solving the integral: $\frac{1}{13.5} \int_{3}^{4} (1 + x) \, dx = \frac{1}{13.5} \left[ x + \frac{x^2}{2} \right]_{3}^{4}$

Evaluating at the bounds: [ \frac{1}{13.5} \left[ (4 + \frac{16}{2}) - (3 + \frac{9}{2}) \right] = \frac{1}{13.5} \left[ 12 - 7.5 \right] = \frac{4.5}{13.5} ]

Simplifying: $P(3 < x < 4) = \frac{4.5}{13.5} \approx 0.333$

Final Answers:

1. $P(x < 4) \approx 0.593$
2. $P(3 < x < 4) \approx 0.333$

Would you like more detailed steps or have any questions?

Follow-up Questions:

1. How do we determine if a given function is a valid PDF?
2. What are the properties of continuous probability distributions?
3. Can you explain how normalization works in probability?
4. What is the significance of calculating probabilities over different intervals?
5. How can we interpret the results of $P(x < 4)$ and $P(3 < x < 4)$?

Tip:

Always ensure that the total integral of a PDF over its entire range equals 1 to confirm it is a valid distribution.