CALCULUS 1 HELP PLEASE

Question

f(x) = ln(2 + sin(x)), 0 ≤ x ≤ 2𝜋

Find the interval(s) on which f is concave up. (Enter your answer using interval notation.)

Find the interval(s) on which f is concave down. (Enter your answer using interval notation.)

Find the inflection points of f.

smaller x-value (x, y) = ( , )

larger x-value (x, y) = ( , )

Answer 1

f ( x ) = l n ( 2 + sin x )

f ‘ ( x ) = cos x / ( 2 + sin x )

f “ ( x ) = [ ( 2 + sin x ) ( - sin x ) ]- [cos x ( cos x ) ] / ( 2 + sin x ) ² = 0—

= - 2 sin x - sin ² x - cos ² x = 0

= - 2 sin x = sin ² x + cos ² x ; sin ² x - cos ² x = 1

= - 2 sin x = 1

= sin x = - 1/2

Inflection: x = 7π / 6 , 11π / 6

− 0 + 0 −

———— Ι————— Ι————

0. 7π / 6. 11π / 6. 2π

up : ( 7π / 6 , 11π / 6 )