Find the number of solutions to the equation sin((pi sqrt(x))/4)+cos((pi sqrt(2−x))/4)=sqrt 2

perlejatyh8

perlejatyh8

Answered question

2022-11-05

Find the number of solutions to the equation sin ( π x 4 ) + cos ( π 2 x 4 ) = 2

Answer & Explanation

Stella Andrade

Stella Andrade

Beginner2022-11-06Added 19 answers

From the comments, the domain is [ 0 , 2 ]
Let f ( x ) = sin ( π x 4 ) + cos ( π 2 x 4 ) 2
f ( x ) = 1 2 x π 4 cos ( π x 4 ) + 1 2 2 x π 4 sin ( π 4 2 x )
Angles of sine and cosine are varying from 0 to π 2 2 . That means f ( x ) > 0. Thus it crosses x-axis either once or never.
Also, since f(x) is increasing, we only need to check values at domain end points.
f ( 0 ) = cos ( π 2 2 ) 2 < 0
f ( 2 ) = sin ( π 2 2 ) + 1 2
4 > 2 2 π 4 < π 2 2 sin ( π 2 2 ) > 1 2 > 0.7
Thus, f ( 2 ) > 0
Thus, the given equation has one and only one solution.

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