Trig function revisited

Algebra Level pending

Given that w > 0 w>0 , If f ( x ) = sin ⁡ ( ω x + π 4 ) f(x)=\sin(\omega x+\dfrac{\pi}{4}) strictly decreases for x ∈ ( π 2 , π ) x \in (\dfrac{\pi}{2},\pi) , what is the range of ω \omega ?

The range can be expressed as [ l , r ] [l,r] . Submit r − l r-l .


The answer is 0.75.

Alice Smith by Alice Smith , 20, China

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1 solution

The function sin ⁡ ( ω x + π 4 ) \sin (\omega x+\dfrac{π}{4}) decreases strictly in the interval ( π 2 , 3 π 2 ) (\dfrac{π}{2}, \dfrac{3π}{2}) . Hence π 4 π 2 ≤ ω ≤ 5 π 4 π \dfrac{\dfrac{π}{4}}{\dfrac{π}{2}}\leq \omega\leq \dfrac{5π}{4π} . Hence l = 1 2 , r = 5 4 l=\dfrac{1}{2}, r=\dfrac{5}{4} , and r − l = 3 4 = 0.75 r-l=\dfrac{3}{4}=\boxed {0.75}

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