It is a result of Narasimhan and Seshadri that if $V$ is semistable and $W$ is stable then $V \otimes W$ is semistable.

If $E$ has a filtration with associated graded $E_i/ E_{i-1}$, and $F$ has a filtration with associated graded $F_j/ F_{j-1}$, then $E \otimes F$ has a filtration with associated graded $(E_i/ E_{i-1}) \otimes (F_j/ F_{j-1})$. By the previous claim, the associated graded pieces of this filtration are semistable, and we can choose the filtration so that these pieces are in order of increasing slope. Hence it is the Harder-Narasimhan filtration.

Thus the slopes of $E \otimes F$ are the slopes of $E$ plus the slope of $F$, and in particular the maximal slope of $E \otimes F$ is the sum of the maximal slope of $E$ and the maximal slope of $F$.

It is a result of Narasimhan and Seshadri that if $V$ is semistable and $W$ is semistable then $V \otimes W$ is semistable.

If $E$ has a filtration with associated graded $E_i/ E_{i-1}$, and $F$ has a filtration with associated graded $F_j/ F_{j-1}$, then $E \otimes F$ has a filtration with associated graded $(E_i/ E_{i-1}) \otimes (F_j/ F_{j-1})$. By the previous claim, the associated graded pieces of this filtration are semistable, and we can choose the filtration so that these pieces are in order of increasing slope. Hence it is the Harder-Narasimhan filtration.

Thus the slopes of $E \otimes F$ are the slopes of $E$ plus the slope of $F$, and in particular the maximal slope of $E \otimes F$ is the sum of the maximal slope of $E$ and the maximal slope of $F$.