![6)\; a)\; \; y=7x+\frac{5}{x^2}+\frac{6}{x}-\sqrt[7]{x^4}\; \; ,\; \; \; \; (\sqrt[7]{x^4}=x^{\frac{4}{7}})\\\\y'=7+5\cdot (-2)x^{-1}+6\cdot \frac{-1}{x^2}-\frac{4}{7}\cdot x^{-\frac{3}{7}}=7-\frac{10}{x}-\frac{6}{x^2}-\frac{4}{7\sqrt[7]{x^3}}\\\\b)\; \; y=\sqrt[5]{7x^2-x+5}-\frac{3}{(x-5)^4}\\\\y'=\frac{1}{5}\cdot (7x^2-x+5)^{-\frac{4}{5}}\cdot (14x-1)-\frac{-3\cdot 4(x-5)^3}{(x-5)^8}=\\\\=\frac{1}{5}\sqrt[5]{(7x^2-x+5)^4}+\frac{12}{(x-5)^5} \\\\c)\; \; y=ctg3x\cdot arccos(3x^2)](https://tex.z-dn.net/?f=6%29%5C%3B%20a%29%5C%3B%20%5C%3B%20y%3D7x%2B%5Cfrac%7B5%7D%7Bx%5E2%7D%2B%5Cfrac%7B6%7D%7Bx%7D-%5Csqrt%5B7%5D%7Bx%5E4%7D%5C%3B%20%5C%3B%20%2C%5C%3B%20%5C%3B%20%5C%3B%20%5C%3B%20%28%5Csqrt%5B7%5D%7Bx%5E4%7D%3Dx%5E%7B%5Cfrac%7B4%7D%7B7%7D%7D%29%5C%5C%5C%5Cy%27%3D7%2B5%5Ccdot%20%28-2%29x%5E%7B-1%7D%2B6%5Ccdot%20%5Cfrac%7B-1%7D%7Bx%5E2%7D-%5Cfrac%7B4%7D%7B7%7D%5Ccdot%20x%5E%7B-%5Cfrac%7B3%7D%7B7%7D%7D%3D7-%5Cfrac%7B10%7D%7Bx%7D-%5Cfrac%7B6%7D%7Bx%5E2%7D-%5Cfrac%7B4%7D%7B7%5Csqrt%5B7%5D%7Bx%5E3%7D%7D%5C%5C%5C%5Cb%29%5C%3B%20%5C%3B%20y%3D%5Csqrt%5B5%5D%7B7x%5E2-x%2B5%7D-%5Cfrac%7B3%7D%7B%28x-5%29%5E4%7D%5C%5C%5C%5Cy%27%3D%5Cfrac%7B1%7D%7B5%7D%5Ccdot%20%287x%5E2-x%2B5%29%5E%7B-%5Cfrac%7B4%7D%7B5%7D%7D%5Ccdot%20%2814x-1%29-%5Cfrac%7B-3%5Ccdot%204%28x-5%29%5E3%7D%7B%28x-5%29%5E8%7D%3D%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B5%7D%5Csqrt%5B5%5D%7B%287x%5E2-x%2B5%29%5E4%7D%2B%5Cfrac%7B12%7D%7B%28x-5%29%5E5%7D%20%5C%5C%5C%5Cc%29%5C%3B%20%5C%3B%20y%3Dctg3x%5Ccdot%20arccos%283x%5E2%29 "6)\; a)\; \; y=7x+\frac{5}{x^2}+\frac{6}{x}-\sqrt[7]{x^4}\; \; ,\; \; \; \; (\sqrt[7]{x^4}=x^{\frac{4}{7}})\\\\y'=7+5\cdot (-2)x^{-1}+6\cdot \frac{-1}{x^2}-\frac{4}{7}\cdot x^{-\frac{3}{7}}=7-\frac{10}{x}-\frac{6}{x^2}-\frac{4}{7\sqrt[7]{x^3}}\\\\b)\; \; y=\sqrt[5]{7x^2-x+5}-\frac{3}{(x-5)^4}\\\\y'=\frac{1}{5}\cdot (7x^2-x+5)^{-\frac{4}{5}}\cdot (14x-1)-\frac{-3\cdot 4(x-5)^3}{(x-5)^8}=\\\\=\frac{1}{5}\sqrt[5]{(7x^2-x+5)^4}+\frac{12}{(x-5)^5} \\\\c)\; \; y=ctg3x\cdot arccos(3x^2)")
0\; \; \; \to \; \; est'\; \; extremum\\\\A=z''_{xx}(M)=2>0\; \; \; \to \; \; min\\\\z_{min}=z(M)=z(-1,1)=(-1)^2-1\cdot 1+1^2-1-1+5=4" alt="7)\; \; z=x^2+xy+y^2+x-y+5\\\\z'_{x}=2x+y+1=0\; ,\; \; z'_{y}=x+2y-1=0\; \; ,\; \; \left \{ {{2x+y=-1} \atop {x+2y=1\, |\cdot (-2)}} \right. \oplus \\\\\left \{ {{2x+y=-1} \atop {-3y=-3}} \right. \; \left \{ {{x=-1} \atop {y=1}} \right. \; \; \Rightarrow \; \; \; M(-1,1)\\\\z''_{xx}=2\; \; ,\; \; z''_{yy}=2\; \; ,\; \; z''_{xy}=1\\\\A=z''_{xx}(M)=2\; \; ,\; \; B=z'_{yy}(M)=2\; \; ,\; \; C=z''_{xy}(M)=1\\\\\Delta =AB-C^2=2\cdot 2-1=3>0\; \; \; \to \; \; est'\; \; extremum\\\\A=z''_{xx}(M)=2>0\; \; \; \to \; \; min\\\\z_{min}=z(M)=z(-1,1)=(-1)^2-1\cdot 1+1^2-1-1+5=4" align="absmiddle" class="latex-formula">