Let $P(x),\ Q(x)$ be polynomials such that :
\[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\]
Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.
Given the polynomials \( P(x) \) and \( Q(x) \) with the following integrals:
\[
\int_0^2 \{P(x)\}^2 \, dx = 14, \quad \int_0^2 P(x) \, dx = 4, \quad \int_0^2 \{Q(x)\}^2 \, dx = 26, \quad \int_0^2 Q(x) \, dx = 2,
\]
we need to find the maximum and minimum values of \( \int_0^2 P(x)Q(x) \, dx \).

1. **Define the integrals and constants:**
   Let \( I = [0, 2] \), so the measure \( \ell = 2 \). Define:
   \[
   a = \int_0^2 P(x) \, dx = 4, \quad A = \int_0^2 \{P(x)\}^2 \, dx = 14,
   \]
   \[
   b = \int_0^2 Q(x) \, dx = 2, \quad B = \int_0^2 \{Q(x)\}^2 \, dx = 26.
   \]
   Let \( C = \int_0^2 P(x)Q(x) \, dx \).

2. **Apply the Cauchy-Schwarz inequality:**
   By the Cauchy-Schwarz inequality for integrals, we have:
   \[
   \left( \int_I (P(x) - p)(Q(x) - q) \, dx \right)^2 \leq \left( \int_I (P(x) - p)^2 \, dx \right) \left( \int_I (Q(x) - q)^2 \, dx \right),
   \]
   where \( p = \frac{a}{\ell} = \frac{4}{2} = 2 \) and \( q = \frac{b}{\ell} = \frac{2}{2} = 1 \).

3. **Expand the left-hand side:**
   \[
   \left( \int_0^2 (P(x) - 2)(Q(x) - 1) \, dx \right)^2 = \left( C - \frac{ab}{\ell} \right)^2 = \left( C - \frac{4 \cdot 2}{2} \right)^2 = (C - 4)^2.
   \]

4. **Expand the right-hand side:**
   \[
   \int_0^2 (P(x) - 2)^2 \, dx = \int_0^2 P(x)^2 \, dx - 2 \int_0^2 2P(x) \, dx + \int_0^2 4 \, dx = A - 4a + 4\ell = 14 - 4 \cdot 4 + 4 \cdot 2 = 6,
   \]
   \[
   \int_0^2 (Q(x) - 1)^2 \, dx = \int_0^2 Q(x)^2 \, dx - 2 \int_0^2 Q(x) \, dx + \int_0^2 1 \, dx = B - 2b + \ell = 26 - 2 \cdot 2 + 2 = 24.
   \]
   Therefore,
   \[
   \left( \int_0^2 (P(x) - 2)^2 \, dx \right) \left( \int_0^2 (Q(x) - 1)^2 \, dx \right) = 6 \cdot 24 = 144.
   \]

5. **Combine the results:**
   \[
   (C - 4)^2 \leq 144 \implies |C - 4| \leq 12 \implies -12 \leq C - 4 \leq 12 \implies -8 \leq C \leq 16.
   \]

6. **Conclusion:**
   The maximum value of \( \int_0^2 P(x)Q(x) \, dx \) is 16, and the minimum value is -8.

The final answer is \(\boxed{[-8, 16]}\).