document.write( "Question 82930: Please, find the maximum extrema for f(x) = cos(x)(cos(x) + 1) using derivatives. \n" ); document.write( "

Algebra.Com's Answer #59515 by Nate(3500) $\"\"$ $\"About$

You can put this solution on YOUR website!
f(x) = cos(x)(cos(x) + 1)
\n" ); document.write( "f'(x) = (cos(x))'(cos(x) + 1) + (cos(x))(cos(x) + 1)'
\n" ); document.write( "f'(x) = -sin(x)(cos(x) + 1) + cos(x)(-sin(x))
\n" ); document.write( "f'(x) = -sin(x)cos(x) - sin(x) - cos(x)sin(x)
\n" ); document.write( "f'(x) = -sin(x)(2cos(x) + 1)
\n" ); document.write( "Slope of tangent (defined by f'(x)) is zero at max and min values.
\n" ); document.write( "0 = -sin(x)(2cos(x) + 1)
\n" ); document.write( "sin(x) = 0 when x = 0, $\"pi\"$ , 2 $\"pi\"$ , 3 $\"pi\"$ ...
\n" ); document.write( "cos(x) = -1/2 when x = 2 $\"pi\"$ /3, 4 $\"pi\"$ /3 ...
\n" ); document.write( "No worries, functions of cosine and sine occillate here.
\n" ); document.write( "f(x) = cos(x)^2 + cos(x)
\n" ); document.write( "f(0) = 1^2 + 1 = 2
\n" ); document.write( "f( $\"pi\"$ ) = (-1)^2 - 1 = 0
\n" ); document.write( "f(2 $\"pi\"$ ) = 1^2 + 1 = 2
\n" ); document.write( "f(2 $\"pi\"$ /3) = (-0.5)^2 - 0.5 = -0.25
\n" ); document.write( "f(4 $\"pi\"$ /3) = (-0.5)^2 - 0.5 = -0.25
\n" ); document.write( "Maximum for f(x) is 2.
\n" ); document.write( "~ Proof ~
\n" ); document.write( " $\"graph%28300%2C300%2C-6.3%2C6.3%2C-4%2C4%2Ccos%28x%29%5E2+%2B+cos%28x%29%2C2%29\"$ \n" ); document.write( "

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