document.write( "Question 1192565: Find the stationary point on the curve y = sinx-0.5x for o < x < π and justify that it is a local maximum. \n" ); document.write( "

Algebra.Com's Answer #824466 by ikleyn(52810) $\"\"$ $\"About$

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\n" ); document.write( "Find the stationary point on the curve y = sinx-0.5x for o < x < π and justify that it is a local maximum.
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document.write( "You are given a function  y = sin(x) - 0.5x.\r\n" );
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document.write( "Its stationary point is the point x, where y'(x) = 0,  or\r\n" );
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document.write( "    y'(x) = cos(x) - 0.5 = 0,\r\n" );
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document.write( "            cos(x) = 0.5.\r\n" );
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document.write( "In the given interval  0 < x < π,  there is only one such point: it is the point  x = 60 degrees,  or  x =  radians\r\n" );
document.write( "(the value about   = 1.046...).\r\n" );
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document.write( "See the plot of the function in the Figure below\r\n" );
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document.write( "                 Plot  y = sin(x) - 0.5x\r\n" );
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document.write( "From the plot, you see that the curve in vicinity of  x =  = 1.046...  is  like a parabola opened down.\r\n" );
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document.write( "The second derivative  y''(x) = -sin(x)  has negative value  y''(pi/3) =  =   at this point.\r\n" );
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document.write( "It justifies that the stationary point  x=   is the local maximum of the function y = sin(x) - 0.5x.\r\n" );
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