Norm of an integral

Question

I'm trying to calculate the infinite norm of $\phi(f)=\int_{0}^{1}f(x)x(1-x) \mathrm{d}x$ with $f\colon[0;1] \to \mathbb{R}$ continuous.

But i don't know where to start.

Answer 1

$\forall f \in C([0,1])$:

$$|\phi(f)| \le \int_0^1 |f(x)| x(1-x)dx \le \|f\|_{\sup} \int_0^1 (x-x^2)dx = \frac16 \|f\|_{\sup}$$

So $\|\phi\| \le 1/6$.

And for $f: [0,1] \to \Bbb R$ defined by $f(x) = 1$ (i.e. the constant function $1$), $\phi(f) = 1/6$, so $\|\phi\| = 1/6$.