Thursday, 7 November 2013

real analysis - Prove that, cosxge12xoverpiforallxin[0,piover2]




By drawing the graph of y=cosx and y=12xπ on [0,π2], it looks obvious due to convex nature of cosine function, which follows-
Graph of cos x and 1-2x/pi
But how to prove it rigorously, using Taylor Series expansion or integration or by any other analysis.
Can anybody prove it?
Thanks for assistance in advance.


Answer



Let f(x)=cos(x)+2πx. Then f(x)=2πsin(x) and, since 2π(0,1) and sin is increasing in [0,π2], there is one and only one x0[0,π2] such that sin(x0)=2π. So, f is increasing on [0,x0] and decreasing on [x0,π2]. But then, since f(0)=f(π2)=1, you have(x[0,π2]):f(x)which is exactly what you want to prove.


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