Completeness of normed linear space $C^1[0,1]$

Question

The Space $C^1[0,1]$ of continuously differentiable function is complete with respect to norm ?

$1.$ $\|f\|=sup_{[0,1]}|f’(x)|$.

$2.$ $\|f\|=sup_{[0,1]}|f(x)|$.

$3.$ $\|f\|=sup_{[0,1]}|f(x)|+sup_{[0,1]}|f’(x)|$.

$4.$ $\|f\|=sup_{[0,1]}|f’(x)|+|f(0)|$.

Option $3$ is standard one is correct option . For option $2$ I know counterexample as $<\sqrt{(x-1/2)^2+1/n}>$ so incorrect option . For option $1$ I know a result that if sequence of derivatives is uniformly convergent and sequence itself is convergent at at least one point then limit is also differentiable . For me it seems that last option is also correct one . For option first it seems that it’s not true but I don’t have counterexample. Please help . Thank you in advance.

Answer 1

The first one is not even a norm since constant functions all have norm $0$.

For 4) use the following: $f_n(x)=f_n(0)+\int_0^{x} f_n'(t)dt$.