Ta có: $y=sin^2x-14sinx.cosx-5coss^2x+3\sqrt[3]{33}$$=\frac{1-cos2x}{2}-7sin2x-5(\frac{1+cos2x}{2})+3\sqrt[3]{33}$
$=-7sin2x-3coss2x+3\sqrt[3]{33}-2$
Theo bất đẳng thức Bunhiacopxki ta có:
$|-7sin2x-3coss2x| \le \sqrt{[(-7)^2+(-3)^2](sin^22x+cos^22x)}=\sqrt{58}$
=> $-\sqrt{58} \le -7sin2x-3cos2x \le \sqrt{58}$
=> $min y=-\sqrt{58}+3\sqrt[3]{33}-2$
Ta CM $min y >0$
<=> $3\sqrt[3]{33}>\sqrt{58}+2$
<=> $891>70\sqrt{58}+356$
<=> $535>70\sqrt{58}$
=> $286225>284200$ (lđ)
=> $miny>0$
=> y chỉ nhận giá trị dương