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kimxakiem2507

[TEX]y=sin^{2010}x+cos^{2010}x=t^{1005}+(1-t)^{1005}\ \ (t=sin^2x,t\in{[0,1])[/TEX]
[TEX]y^'=1005[t^{1004}-(1-t)^{1004}][/TEX][TEX],\ \ y^'=0\Leftrightarrow{t^{1004}=(1-t)^{1004}(1)[/TEX]
[TEX]f(x)=x^{1004}\ \ (x\in{[0,1])[/TEX]
[TEX]f^'(x)=1004x^{1003}\ge{0}\ \ \forall{x\in{[0,1][/TEX]
[TEX](1)\Leftrightarrow{f(t)=f(1-t)\Leftrightarrow{t=1-t\Leftrightarrow{t=\frac{1}{2}[/TEX]
[TEX]BBT\Rightarrow{y_{max}=1[/TEX][TEX]\ \ \left[t=0\\t=1\Leftrightarrow{x=k\frac{\pi}{2}\ \ (k\in{Z)[/TEX]
[TEX]y_{min}=\frac{1}{2^{1004}}[/TEX][TEX]\ \ t=\frac{1}{2}\Leftrightarrow{x=\frac{\pi}{4}+k \frac{\pi}{2}\ \ (k\in{Z)[/TEX]
 
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