document.write( "Question 1192776: Determine the value of x , where 0 ≤ x ≤ π , for which the curve y = 2cosx + 3sinx has a stationary point and determine the nature of this point. \n" ); document.write( "

Algebra.Com's Answer #824676 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( " $\"f%28x%29=2cos%28x%29%2B3sin%28x%29\"$

\n" ); document.write( " $\"df%2Fdx=-2sin%28x%29%2B3cos%28x%29\"$

\n" ); document.write( " $\"d%5E2f%2Fdx%5E2=-2cos%28x%29-3sin%28x%29\"$

\n" ); document.write( "The stationary point is where the derivative is zero.

\n" ); document.write( " $\"-2sin%28x%29%2B3cos%28x%29=0\"$
\n" ); document.write( " $\"2sin%28x%29=3cos%28x%29\"$
\n" ); document.write( " $\"sin%28x%29%2Fcos%28x%29=3%2F2\"$
\n" ); document.write( " $\"x+=+tan%5E%28-1%29%283%2F2%29\"$ ~ 0.983 radians

\n" ); document.write( "There is little value in evaluating the function by hand when x has the value arctan(3/2); use a graphing calculator to find the approximate y value at the stationary point.

\n" ); document.write( "The nature of the stationary point is determined by the sign of the second derivative. The graphing calculator will show that the stationary point is a maximum.

\n" ); document.write( "However, you can tell that the stationary point will be a maximum by looking at the second derivative of the function. The stationary point is when x is between 0 and pi/2, so we are in the first quadrant, where sin(x) and cos(x) are both positive; so the sign of the second derivative is clearly negative, which means the stationary point is a maximum.

\n" ); document.write( "A graph, for x from -pi/2 to 3pi/2, showing a maximum at approximately (1,3.6)....

\n" ); document.write( " $\"graph%28400%2C400%2C-pi%2F2%2C3pi%2F2%2C-5%2C5%2C2cos%28x%29%2B3sin%28x%29%29\"$

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