How do you find the local maximum and minimum values of #f(x) = 2x^3  5x +1# in the the interval is (3,3)?
graph{2x^35x+1 [10, 10, 5, 5]}
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To find the local maximum and minimum values of ( f(x) = 2x^3  5x + 1 ) in the interval ((3, 3)), follow these steps:

Find the critical points by setting the derivative of ( f(x) ) equal to zero and solving for ( x ).
( f'(x) = 6x^2  5 )
Set ( f'(x) = 0 ) and solve for ( x ):
( 6x^2  5 = 0 )
( 6x^2 = 5 )
( x^2 = \frac{5}{6} )
( x = \pm \sqrt{\frac{5}{6}} )
The critical points are ( x = \sqrt{\frac{5}{6}} ) and ( x = \sqrt{\frac{5}{6}} ).

Evaluate ( f(x) ) at the critical points and at the endpoints of the interval ((3, 3)).
( f(\sqrt{\frac{5}{6}}) \approx 4.63 )
( f(\sqrt{\frac{5}{6}}) \approx 2.63 )
( f(3) = 47 )
( f(3) = 52 )

Compare the values obtained in step 2 to determine which one is the maximum and which one is the minimum within the given interval.
The local maximum value is ( f(3) = 52 ).
The local minimum value is ( f(\sqrt{\frac{5}{6}}) \approx 4.63 ).
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