The machine M produces a product P. From the two years I and II it is known that the unit costs for manufacturing the product in the first year are kI  €42.50 and in the second year kII  €41.00. In year I 2 000 units were produced and in year II 2 500 units were produced. a) What are the values of m and b if the total costs K result from the quantity x according to the equation K(x)  m · x  b and this applies to both years I and II? State the values for m and b as well as the cost function with the values for m and b.

Given that the total costs are represented by the equation K(x) = m \cdot x + b, where x is the quantity produced, m is the unit cost, and b is a constant term.

For year I, the total cost is given by K_I(x) = k_I \cdot 2000 = 42.50 \cdot 2000 = 85,000 euros.
For year II, the total cost is given by K_{II}(x) = k_{II} \cdot 2500 = 41.00 \cdot 2500 = 102,500 euros.

Since the equation K(x) = m \cdot x + b applies to both years I and II:
1. For year I: K_I(x) = m \cdot 2000 + b
2. For year II: K_{II}(x) = m \cdot 2500 + b

By substituting the known values:
1. For year I: 85,000 = 2000m + b
2. For year II: 102,500 = 2500m + b

Solving these two equations simultaneously to find the values of m and b:
From equation 1: b = 85,000 - 2000m
Substitute b into equation 2: 102,500 = 2500m + 85,000 - 2000m
102,500 = 2500m + 85,000 - 2000m
102,500 = 500m + 85,000
500m = 17,500
m = 35

Substitute m = 35 back into b = 85,000 - 2000m:
b = 85,000 - 2000 \times 35
b = 85,000 - 70,000
b = 15,000

Therefore, the values for m and b are:
m = 35
b = 15,000

The cost function with the values for m and b is:
K(x) = 35x + 15,000 euros.

\boxed{m = 35, \ b = 15,000, \ K(x) = 35x + 15,000}