Is it correct to say that `K_(max)` is proportional to f ? If not what would a correct statement of the relationship between `K_(max)` and f?

Explanation

To determine the relationship between the maximum kinetic energy \( K_{max} \) of photoelectrons and the frequency \( f \) of the incident light, we start with the photoelectric effect equation.

### Step-by-Step Solution:

1. **Understand the Photoelectric Effect**: The photoelectric effect states that when light of sufficient frequency (or energy) strikes a material, it can eject electrons from that material. The maximum kinetic energy of these ejected electrons is given by the equation:
\[ K_{max} = E - W \]
where \( E \) is the energy of the incident photon and \( W \) is the work function of the material.

2. **Photon Energy**: The energy \( E \) of a photon can be expressed in terms of its frequency \( f \):
\[ E = hf \]
where \( h \) is Planck's constant.

3. **Substituting Photon Energy into the Kinetic Energy Equation**: We can substitute the expression for photon energy into the kinetic energy equation:
\[ K_{max} = hf - W \]

4. **Expressing Work Function in Terms of Frequency**: The work function \( W \) can also be expressed in terms of the threshold frequency \( f_0 \):
\[ W = hf_0 \]
where \( f_0 \) is the minimum frequency required to eject an electron.

5. **Final Expression for Maximum Kinetic Energy**: Substituting this into the equation for \( K_{max} \):
\[ K_{max} = hf - hf_0 \]
This simplifies to:
\[ K_{max} = h(f - f_0) \]

6. **Analyzing the Relationship**: From the final expression, we can see that:
\[ K_{max} \propto (f - f_0) \]
This indicates that \( K_{max} \) is directly proportional to the difference between the frequency of the incident light \( f \) and the threshold frequency \( f_0 \).

### Conclusion:
It is not correct to say that \( K_{max} \) is directly proportional to \( f \). The correct statement is that \( K_{max} \) is directly proportional to the difference \( (f - f_0) \).