Find the greatest values of sin (x − π /3 ) sin ( π/6 + x) and also find the corresponding value of x for which it is greatest, where x ∈ [π, 3π /2 ].
Answers
Given :
sin (x − π /3 ) sin ( π/6 + x)
To Find : greatest values
corresponding value of x for which it is greatest, where x ∈ [π/2, 3π /2 ].
Solution:
f(x) = sin (x − π /3 ) sin ( π/6 + x)
f'(x) = sin (x − π /3) Cos ( π/6 + x) + cos (x − π /3 ) sin ( π/6 + x)
= Sin(x - π /3 + π/6 + x )
= Sin( 2x - π/6)
f'(x) = 0
Sin( 2x - π/6) = 0
=> x = π/12 , 7π/12 , 13π/12 , 19π/12
7π/12 , 13π/12 ∈ [π/2, 3π /2 ]
f''(x) = Cos ( 2x - π/6)
x = 7π/12 f''(x) is -ve
Hence max value at x = 7π/12
f(7π/12) = sin (7π/12 − π /3 ) sin ( π/6 + 7π/12)
= sin( 3π/12) sin (9π/12)
= sin( π/4) sin (3π/4)
= (1/√2) (1/√2)
= 1/2
Max value = 1/2 at x = 7π/12
if Question is x ∈ ∈ [π , 3π /2 ]
then at 3π /2 is max value
sin (3π/2 − π /3 ) sin ( π/6 + 3π/2)
= sin( 7π/6)sin(5π/3)
= √3/4
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