application of derivatives

1 Answers

Consider f(x) = x - Ï€3sinx

f(0) = f(Ï€6) = 0

f '(x) = 1 - Ï€3cosx

f '(x) = 0 only at one point between 0 and Ï€6 given by cosx = 3Ï€

so between 0 and Ï€6 the curve is either above x-axis or below x-axis

f ''(x) = Ï€3sinx > 0, so the curve is concave upwards which means it lies below x-axis

so, f(x) < 0

x - Ï€3sinx < 0

x(cosecx) < Ï€3

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