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Let f(x),xgeq0, be a non-negative contin...

Let `f(x),xgeq0,` be a non-negative continuous function. If `f^(prime)(x)cosxlt=f(x)sinxAAxgeq0,` then find `f((5pi)/3)`

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Given `f(x)ge 0 forall x ge 0`
`f(x) cos x-f(x) sinx le0`
`f(x)cosxle0`
g(x) =f(x)cosx
`g(x)le0`
g(x) is a decreasing function.
Now `f(5pi)/(3)ge0,(as f(x)geforallxge0)`
`g(5pi)/(3)le0` ltbr. `f(5pi)/(3)le0`
From statement (2) and (3) ,`f(5pi)/(3)=0`
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