js","1717","static/chunks/1717-e9687bfca52045a6.js","8117","static/chunks/8117-3a5cb76430604345.js","1464","static/chunks/1464-4de8faf93b802138.js","422","static/chunks/422-b1f5ec985dcfd2ef.js","3169","static/chunks/3169-0b225cc6264f1f70.js","5791","static/chunks/5791-b7519a5ea5ac96a7.js","4839","static/chunks/4839-abf7392d0b162b93.js","5322","static/chunks/5322-2dee03e10531d285.js","1493","static/chunks/1493-0e365f06d2d7ff74.js","3910","static/chunks/3910-d35f7a55e63a44d2.js","2661","static/chunks/2661-5a8e341dbac91d3f.js","6986","static/chunks/app/(unauthorized-user)/testing/%5BexamType%5D/%5B...slug%5D/page-3d61388e5ed47f00.js"],""] 46:T690, $9^{4x-x^2-1}-36\cdot 3^{4x-x^2-1}+243\ge0 \quad \Leftrightarrow \quad 3^{2(4x-x^2-1)}-36\cdot 3^{4x-x^2-1}+243\ge0.$

Пусть $\; 3^{4x-x^2-1}=t. \;$ Тогда неравенство примет вид: $\quad t^2-36t+243\ge0.$

$t^2-36t+243=0 \quad \Leftrightarrow \quad \begin{cases} t=27,\\ t=9. \end{cases}$ $t^2-36t+243\ge0 \quad \Leftrightarrow \quad (t-27)(t-9)\ge0.$

Решим полученное неравенство методом интервалов:

$\begin{cases} t\le 9,\\ t\ge 27 \end{cases} \quad \Leftrightarrow \quad \begin{cases} 3^{4x-x^2-1}\le 3^2,\\ 3^{4x-x^2-1}\ge 3^3 \end{cases} \quad \Leftrightarrow \quad \begin{cases} 4x-x^2-1\le2,\\ 4x-x^2-1\ge3. \end{cases}$ $\Leftarrow \quad \begin{cases} x^2-4x+3\ge0,\\ x^2-4x+4\le0 \end{cases} \quad \Leftrightarrow \quad \begin{cases} (x-3)(x-1)\ge0,\\ (x-2)^2\le0 \end{cases} \quad \Leftrightarrow \quad \begin{cases} x\le1,\; x\ge3,\\ x=2. \end{cases}$

Задание №29384 ЕГЭ по Математике