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Let's analyze the problem step by step.

The resistance \( R \) is given as:

\[
R = \frac{KL(\gamma + 1)^2}{D^4}
\]

where:

- \( l \) is the length of the vessel,
- \( D \) is the diameter,
- \( \gamma \geq 0 \) is the curvature,
- \( K > 0 \) represents the viscosity of the blood.

You are asked to determine whether the following statements are true or false:

### (i) \( R \) is decreasing
- To determine whether \( R \) is decreasing with respect to \( D \), we can inspect the form of the equation. Since \( R \propto \frac{1}{D^4} \), it is clear that as \( D \) increases, \( R \) decreases because of the negative power. So, statement (i) is **True**.

### (ii) \( R \) has no critical numbers
- Critical numbers occur where \( \frac{dR}{dD} = 0 \) or is undefined. From the formula for \( R \), we observe that it is smooth and differentiable for \( D > 0 \). Let's differentiate \( R \) with respect to \( D \).

The derivative is:

\[
\frac{dR}{dD} = -4 \frac{KL(\gamma + 1)^2}{D^5}
\]

This is never zero since \( KL(\gamma + 1)^2 > 0 \) for all \( D > 0 \). Therefore, statement (ii) is **True** since \( R \) has no critical numbers.

### (iii) \( \frac{dR}{dD} \) is decreasing
- From the derivative \( \frac{dR}{dD} = -4 \frac{KL(\gamma + 1)^2}{D^5} \), observe that the derivative has a negative power of \( D \). As \( D \) increases, the magnitude of \( \frac{dR}{dD} \) decreases because \( D^5 \) increases, making the overall value less negative. Hence, statement (iii) is **False** since \( \frac{dR}{dD} \) is not decreasing but rather increasing in magnitude (becoming less negative).

### Final answers:
- (i) True (1)
- (ii) True (1)
- (iii) False (2)

The correct response would be **1,1,2**.

Would you like further details on the calculations, or do you have any other questions?

Here are 5 related questions for deeper understanding:

1. What does it mean for a function to be "decreasing" in mathematical terms?
2. How do you find critical points of a function?
3. Why is the derivative \( \frac{dR}{dD} \) always negative for the given function?
4. How would the resistance change if \( D \) were to double?
5. What is the significance of higher-order derivatives in this context?

**Tip:** To determine whether a function is increasing or decreasing, always examine the sign of its first derivative.

The resistance R of the blood flow through a vessel is given by R = K L (γ + 1)^2 / D^4, where l is the length of the vessel, D is its diameter, γ ≥ 0 is the curvature, and K > 0 represents the viscosity of the blood. Consider the following statements: (i) R is decreasing, (ii) R has no critical numbers, (iii) dR/dD is decreasing. Determine which of the above statements are True (1) or False (2).

Explore the mathematical analysis of the resistance R of blood flow in a vessel. Learn how to determine if R is decreasing, identify critical points, and evaluate the behavior of dR/dD. This problem involves derivatives, exponential and rational functions, and critical points. Suitable for advanced students in Grades 11-12.

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