[tex]\displaystyle\bf\\3Cos^{2} \alpha -4Sin^{2} \alpha =3\cdot(1-Sin^{2}\alpha )-4Sin^{2} \alpha =3-3Sin^{2} \alpha -4Sin^{2} \alpha=\\\\\\=3-7Sin^{2} \alpha =3-7\cdot\frac{1-Cos2\alpha }{2} =3-3,5\cdot(1-Cos2\alpha )=\\\\\\=3-3,5+3,5Cos2\alpha =-0,5+3,5Cos2\alpha\\\\\\-1\leq Cos2\alpha \leq 1\\\\\\-1\cdot 3,5\leq 3,5\cdot Cos2\alpha \leq 1\cdot3,5\\\\\\-3,5\leq 3,5Cos2\alpha \leq 3,5\\\\\\-0,5-3,5\leq -0,5+3,5Cos2\alpha \leq -0,5+3,5\\\\\\-4\leq -0,5+3,5Cos2\alpha \leq 3[/tex]

Ответ : наибольшее значение выражения равно 3 .