Given $f \in C^{3}[0,2]$, we approximate $f^{\prime}(0)$ by the linear combination

$$\mu(f)=-\frac{3}{2} f(0)+2 f(1)-\frac{1}{2} f(2)$$

Using the Peano kernel theorem, determine the least constant $c$ in the inequality

$$\left|f^{\prime}(0)-\mu(f)\right| \leq c\left\|f^{\prime \prime \prime}\right\|_{\infty},$$

and give an example of $f$ for which the inequality turns into equality.