![1.\;\frac{n^3-nm^2}{mn-n^2}=\frac{n(n^2-m^2)}{n(m-n)}=-\frac{(m-n)(m+n)}{m-n}=-m-n\\\\2.\;\log_x125=3\Rightarrow x^3=125\Rightarrow x=5\\\log_2y=2\Rightarrow y=2^2=4\\\sqrt{x+y}=\sqrt{5+4}=\sqrt9=\pm3\\\\3.\;1-\sin^2\alpha-\cos^2\alpha=1-\left(\sin^2\apha+\cos^2\alpha\right)=1-1=0\\\\4.\;\sqrt[4]{\sqrt[3]{5^x}}=125\\\left(\sqrt[3]{5^x}\right)^{\frac14}=5^3\\\left(5^x\right)^{\frac1{12}}=5^3\\5^{\frac x{12}}=5^3\\\frac x{12}=3\\x=36](https://tex.z-dn.net/?f=1.%5C%3B%5Cfrac%7Bn%5E3-nm%5E2%7D%7Bmn-n%5E2%7D%3D%5Cfrac%7Bn%28n%5E2-m%5E2%29%7D%7Bn%28m-n%29%7D%3D-%5Cfrac%7B%28m-n%29%28m%2Bn%29%7D%7Bm-n%7D%3D-m-n%5C%5C%5C%5C2.%5C%3B%5Clog_x125%3D3%5CRightarrow+x%5E3%3D125%5CRightarrow+x%3D5%5C%5C%5Clog_2y%3D2%5CRightarrow+y%3D2%5E2%3D4%5C%5C%5Csqrt%7Bx%2By%7D%3D%5Csqrt%7B5%2B4%7D%3D%5Csqrt9%3D%5Cpm3%5C%5C%5C%5C3.%5C%3B1-%5Csin%5E2%5Calpha-%5Ccos%5E2%5Calpha%3D1-%5Cleft%28%5Csin%5E2%5Capha%2B%5Ccos%5E2%5Calpha%5Cright%29%3D1-1%3D0%5C%5C%5C%5C4.%5C%3B%5Csqrt%5B4%5D%7B%5Csqrt%5B3%5D%7B5%5Ex%7D%7D%3D125%5C%5C%5Cleft%28%5Csqrt%5B3%5D%7B5%5Ex%7D%5Cright%29%5E%7B%5Cfrac14%7D%3D5%5E3%5C%5C%5Cleft%285%5Ex%5Cright%29%5E%7B%5Cfrac1%7B12%7D%7D%3D5%5E3%5C%5C5%5E%7B%5Cfrac+x%7B12%7D%7D%3D5%5E3%5C%5C%5Cfrac+x%7B12%7D%3D3%5C%5Cx%3D36 "1.\;\frac{n^3-nm^2}{mn-n^2}=\frac{n(n^2-m^2)}{n(m-n)}=-\frac{(m-n)(m+n)}{m-n}=-m-n\\\\2.\;\log_x125=3\Rightarrow x^3=125\Rightarrow x=5\\\log_2y=2\Rightarrow y=2^2=4\\\sqrt{x+y}=\sqrt{5+4}=\sqrt9=\pm3\\\\3.\;1-\sin^2\alpha-\cos^2\alpha=1-\left(\sin^2\apha+\cos^2\alpha\right)=1-1=0\\\\4.\;\sqrt[4]{\sqrt[3]{5^x}}=125\\\left(\sqrt[3]{5^x}\right)^{\frac14}=5^3\\\left(5^x\right)^{\frac1{12}}=5^3\\5^{\frac x{12}}=5^3\\\frac x{12}=3\\x=36")
0\\x\in(-2;\;0)\Rightarrow y'<0\\x\in(0;\;2)\Rightarrow y'<0\\x\in(2;\;+\infty)\Rightarrow y>0" alt="5.\;f(x)=x+\frac4x\\f'(x)=1-\frac4{x^2}\\1-\frac4{x^2}=0\\\frac4{x^2}=1\\x^2=4\\x_1=-2,\;x_2=2\\x\in(-\infty;\;-2)\Rightarrow y'>0\\x\in(-2;\;0)\Rightarrow y'<0\\x\in(0;\;2)\Rightarrow y'<0\\x\in(2;\;+\infty)\Rightarrow y>0" align="absmiddle" class="latex-formula">
В точке -2 производная меняет знак с "плюса" на "минус", значит в этой точке функция имеет максимум.