which of he following is a metric space? $a)$ $C^1[0,1]$ of continuously differentiable real valued functions on $[0,1]$ with the metric $$d(f,g)=\max_{t\in[0,1]}|f-g|$$
I am sure that it is not complete, but could any one help me to construct a counter example? well, $f(x)=|x-1/2|$ will work?
$b)$ The space of all polynomial with real coeffi single variable with same metric as above, this is also not complete, $|x-1/2|$ will work?

For part (a), your function $f$ is not differentiable at $x=1/2$, and so $f\not\in C^1[0,1]$. 
To get a counterexample, try to find a limiting function $f\in C^0[0,1]\setminus C^1[0,1]$ (i.e. a continuous, non-differentiable function) and a sequence of smooth functions that converges to $f$ in $C^0[0,1]$. This convergent (in $C^0$) sequence can be chosen to give you a Cauchy sequence in $C^1[0,1]$ with the metric you have indicated - but the limit 'falls out' of $C^1$, giving the counterexample you need. Indeed your suggested $f$ could be used as the limiting function. For the sequence, apply a mollifier to $f$.
Similarly for (b), your suggested $f$ is not in the space in question.
