Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq

Question

For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm?

-I can easily show the supremum of the norm of a function continuous once is positive and homogeneous but im not sure how to show the triangle inequality holds and have no clue why it is not a norm.

Any guidance would be appreciated