For $x\geq 0,$ find the minimum value of $x$ for which $\int_0^x 2^t(2^t-3)(x-t)\ dt$ is minimized.
To find the minimum value of \( x \geq 0 \) for which the integral 
\[ \int_0^x 2^t(2^t-3)(x-t)\, dt \]
is minimized, we will follow these steps:

1. **Define the integral as a function \( G(x) \):**
   \[ G(x) = \int_0^x 2^t(2^t-3)(x-t)\, dt \]

2. **Differentiate \( G(x) \) with respect to \( x \):**
   Using the Leibniz rule for differentiation under the integral sign, we get:
   \[ G'(x) = \frac{d}{dx} \left( \int_0^x 2^t(2^t-3)(x-t)\, dt \right) \]
   \[ G'(x) = 2^x(2^x-3)(x-x) + \int_0^x \frac{\partial}{\partial x} \left( 2^t(2^t-3)(x-t) \right) dt \]
   \[ G'(x) = \int_0^x 2^t(2^t-3) \, dt \]

3. **Simplify the integral:**
   Let \( f(t) = 2^t(2^t-3) \). Then,
   \[ G'(x) = \int_0^x f(t) \, dt \]

4. **Evaluate the integral \( \int_0^x f(t) \, dt \):**
   \[ f(t) = 2^t(2^t-3) = 4^t - 3 \cdot 2^t \]
   \[ \int_0^x f(t) \, dt = \int_0^x (4^t - 3 \cdot 2^t) \, dt \]
   \[ = \int_0^x 4^t \, dt - 3 \int_0^x 2^t \, dt \]
   \[ = \left[ \frac{4^t}{\ln 4} \right]_0^x - 3 \left[ \frac{2^t}{\ln 2} \right]_0^x \]
   \[ = \frac{4^x - 1}{\ln 4} - 3 \cdot \frac{2^x - 1}{\ln 2} \]

5. **Set \( G'(x) = 0 \) and solve for \( x \):**
   \[ \frac{4^x - 1}{\ln 4} - 3 \cdot \frac{2^x - 1}{\ln 2} = 0 \]
   \[ \frac{4^x - 1}{2 \ln 2} - 3 \cdot \frac{2^x - 1}{\ln 2} = 0 \]
   \[ \frac{4^x - 1}{2} - 3(2^x - 1) = 0 \]
   \[ 4^x - 1 - 6 \cdot 2^x + 6 = 0 \]
   \[ 4^x - 6 \cdot 2^x + 5 = 0 \]

6. **Solve the equation \( 4^x - 6 \cdot 2^x + 5 = 0 \):**
   Let \( y = 2^x \). Then the equation becomes:
   \[ y^2 - 6y + 5 = 0 \]
   \[ (y-1)(y-5) = 0 \]
   \[ y = 1 \quad \text{or} \quad y = 5 \]
   Since \( y = 2^x \), we have:
   \[ 2^x = 1 \quad \text{or} \quad 2^x = 5 \]
   \[ x = 0 \quad \text{or} \quad x = \log_2 5 \]

7. **Determine the minimum value:**
   We need to check the values of \( G(x) \) at \( x = 0 \) and \( x = \log_2 5 \):
   \[ G(0) = \int_0^0 2^t(2^t-3)(0-t) \, dt = 0 \]
   \[ G(\log_2 5) = \int_0^{\log_2 5} 2^t(2^t-3)(\log_2 5 - t) \, dt \]
   Since \( G(0) = 0 \) and \( G(x) \) is minimized at \( x = 0 \), the minimum value of \( x \) is \( 0 \).

The final answer is \( \boxed{0} \)