For a homework question I have to solve the following problem:Let $f \in C^2([0,1])$. The task is to show that there exists $M>0$ (independent of $f$) such that$$||f'||_\infty \le M ||f||_\infty (||f||_\infty + ||f''||_\infty)$$where $||f||_\infty$ denotes the supremum norm on the interval $[0,1]$.
Here's what I tried so far:I know that there is a statement (e.g. in Rudin's analysis book) that says $||f'||_\infty^2 \le 4 ||f||_\infty ||f''||_\infty$, which is normally formulated for functions on $\mathbb R$ but works the same way (e.g. by extending the function $f$ to be constant outside $[0,1]$) on an interval. I can obviously use this inequality together with Young's inequality to obtain the estimate $||f'||_\infty \le \sqrt 2 (||f||_\infty + ||f''||_\infty)$ which is even better than the one I need in the case $||f||_\infty \ge 1$.
However, nothing I have tried so far worked for the general case where $||f||_\infty$ could also be smaller than $1$.
I tried to use the mean value theorem for $f$ or the fundamental theorem of calculus for either $f$ or $f'$, but somehow I never ended up with an estimate good enough. Any help would be appreciated.