Let $f$ $\in$ C([0,1]), $f(0)=0$ and $\epsilon > 0$. Prove there exists a polynomial $p$ such that $p(0)=f(0)=0$, $p´(0)=0$ and $||p-f|| < \epsilon$ . The norm is sup-norm
I Know that by Walsh theorem I can have the approximation condition and the interpolation but I don´t have the one about the derivative in $0$, also for another exercise in my class I know I can have an approximating polynomial with every $x^{4k} $ for $k=0, 1, 2, ...n$ where $n$ is the polynomial degree, but in this case I have the derivative condition but not $p(0)=0$.
You say you already know how to show that there exists a polynomial $q$ such that $q'(0)=0$ and $\|q-f\|<\epsilon$. Note that the constant term of $q$ is equal to $q(0)$, and must be smaller than $\epsilon$ since $f(0)=0$. Now just remove the constant term from $q$ to get a polynomial $p$ such that $p(0)=0$ as well, and note that $\|p-q\|<\epsilon$.