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The problem provided involves finding \( U_{yy} \), given the function \( f(z) = z^2 - 3z \).

Let’s analyze the situation:

1. The function \( f(z) = z^2 - 3z \) is given, and typically \( z \) is a variable in terms of \( x \) and \( y \), i.e., \( z = x + y \) or \( z = x - y \), etc., in multivariable calculus.
2. The second derivative with respect to \( y \), \( U_{yy} \), needs to be computed.

However, the problem does not directly specify what \( z \) is in terms of \( y \), but based on the options, it is likely that \( z = y \).

Let’s proceed step-by-step assuming \( z = y \):

### Step 1: Differentiate \( f(z) = z^2 - 3z \) with respect to \( y \).
- First derivative: \( f'(z) = \frac{d}{dy} \left( z^2 - 3z \right) = 2z - 3 \).
- Second derivative: Since \( z = y \), we differentiate again with respect to \( y \):  
  \( f''(z) = \frac{d}{dy} \left( 2z - 3 \right) = 2 \).

### Conclusion:
The correct value of \( U_{yy} \) is \(-2\), which corresponds to option **b**.

Would you like further details on this calculation or related questions?

Here are 5 related questions for practice:
1. If \( f(z) = z^2 - 4z + 1 \), what is \( U_{yy} \) if \( z = y \)?
2. How do you find \( U_{yy} \) if \( f(z) = 3z^3 - 2z^2 + z \)?
3. If \( z = x + y \), how would you find \( U_{yy} \)?
4. What is the second derivative with respect to \( y \) if \( f(z) = \sin(z) \) and \( z = y \)?
5. How does the second derivative relate to the curvature of a function?

**Tip:** When differentiating functions, always check the dependence of each variable carefully before computing derivatives.

This problem involves finding the second derivative with respect to y, U_yy, given the function f(z) = z^2 - 3z. The solution applies key calculus concepts like partial derivatives and the second derivative test.

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