Question

Найдите множество значений функции y= - x^4 - 4x^2 + 5

Answer 1

Ответ:   $y\in (-\infty ;\, 5\ ]$  .

$y=-x^4-4x^2+5\\\\-x^4-4x^2+5=-(x^4+4x^2)+5=-\Big((x^2+2)^2-4\Big)+5=-(x^2+2)^2+9\\\\x^2\geq 0\ \ ,\ \ x^2+2\geq 2\ \ ,\ \ (x^2+2)^2\geq 4\ \ ,\ \ -(x^2+2)^2\leq -4\ \ \Rightarrow \\\\-(x^2+2)^2+9\leq -4+9\\\\-(x^2+2)^2+9\leq 5\\\\y\in (-\infty ;\ 5\ ]$