Let be the set $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left( 
x_1 \right) = f\left( x_2 \right) =0\vee  f\left( 
x_1 \right) = f'\left( x_1 \right) = 0\right)  \wedge f''<1 \right\} , $

and $ f^*\in\mathcal{C} $ such that $ \int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx .  $

Find $ \int_0^1\left| f^*(x) \right| dx $ and describe $ f^*. $
1. **Understanding the Set $\mathcal{C}$:**
   The set $\mathcal{C}$ consists of functions $f:[0,1] \to \mathbb{R}$ that satisfy the following conditions:
   - $f$ has a second derivative $f''$ on $[0,1]$.
   - There exist $x_1, x_2 \in [0,1]$ such that $x_1 \neq x_2$ and either $f(x_1) = f(x_2) = 0$ or $f(x_1) = f'(x_1) = 0$.
   - $f'' < 1$ on $[0,1]$.

2. **Objective:**
   We need to find a function $f^* \in \mathcal{C}$ such that 
   \[
   \int_0^1 |f^*(x)| \, dx = \sup_{f \in \mathcal{C}} \int_0^1 |f(x)| \, dx
   \]
   and describe $f^*$.

3. **Analyzing the Conditions:**
   - The condition $f'' < 1$ implies that the second derivative of $f$ is strictly less than 1. This means that $f$ is concave downwards but not too steep.
   - The condition involving zeros or critical points suggests that $f$ must have specific behavior at certain points.

4. **Constructing a Candidate Function:**
   Consider the function $f(x) = \frac{1}{2}x(1-x)$. This function has the following properties:
   - $f(0) = f(1) = 0$, satisfying the condition of having zeros at distinct points.
   - The second derivative $f''(x) = -1$, which satisfies $f'' < 1$.

5. **Calculating the Integral:**
   \[
   \int_0^1 |f(x)| \, dx = \int_0^1 \left| \frac{1}{2}x(1-x) \right| \, dx = \int_0^1 \frac{1}{2}x(1-x) \, dx
   \]
   Simplifying the integral:
   \[
   \int_0^1 \frac{1}{2}x(1-x) \, dx = \frac{1}{2} \int_0^1 (x - x^2) \, dx
   \]
   \[
   = \frac{1}{2} \left( \int_0^1 x \, dx - \int_0^1 x^2 \, dx \right)
   \]
   \[
   = \frac{1}{2} \left( \left[ \frac{x^2}{2} \right]_0^1 - \left[ \frac{x^3}{3} \right]_0^1 \right)
   \]
   \[
   = \frac{1}{2} \left( \frac{1}{2} - \frac{1}{3} \right)
   \]
   \[
   = \frac{1}{2} \left( \frac{3}{6} - \frac{2}{6} \right)
   \]
   \[
   = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}
   \]

6. **Verifying Optimality:**
   To ensure that $f(x) = \frac{1}{2}x(1-x)$ is indeed the function that maximizes the integral, we need to consider the constraints. Given the constraints on $f''$ and the requirement for zeros or critical points, $f(x) = \frac{1}{2}x(1-x)$ is a reasonable candidate that satisfies all conditions.

Conclusion:
\[
\boxed{\int_0^1 |f^*(x)| \, dx = \frac{1}{12}}
\]