For the function f ( x ) = sin x + 3 x − π 2 ( x 2 + x ) , where x ∈ [ 0 , 2 π ] , consider the following two statements : (I) f is increasing in ( 0 , 2 π ) . (II) f ′ is decreasing in ( 0 , 2

Question

For the function f ( x ) = sin x + 3 x − π 2 ​ ( x 2 + x ) , where x ∈ [ 0 , 2 π ​ ] , consider the following two statements : (I) f is increasing in ( 0 , 2 π ​ ) . (II) f ′ is decreasing in ( 0 , 2 π ​ ) . Between the above two statements,

TestHub · Accepted Answer

f ( x ) = sin x + 3 x − π 2 ( x 2 + x ) x ∈ [ 0 , 2 π ] f ′ ( x ) = cos x + 3 − π 2 ( 2 x + 1 ) > 0 f ( x ) ↑ f ′ ( x ) = − sin x + 0 − 2 π ( 2 ) = − sin x − π 4 < 0 f ′ ( x ) ↓ 0 < x < 2 π ⇒ − π 2 ( + 1 0 < + 1 2 x < + 1 π ) + 3 − π 2 > π − 2 ( + 3 2 x + 1 ) > − π 2 ( + 3 π + 1 ) + v e 3 − π 2 > 3 − π 2 ( 2 x + 1 ) > 3 − π 2 ( + v e π + 1 )

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