A quick way to do this is by recalling the fact that [tex]-1\le\cos x\le1[/tex], so the largest value that [tex]\cos ax[/tex] alone (for any real value of [tex]a[/tex]) can take on is 1.
This means [tex]-3\le3\cos2x\le3[/tex], and so [tex]1\le3\cos2x+4\le7[/tex].
So the maximum value of this function is 7.
