Ex 6.5, 8 - At what points in interval [0, 2pi], does sin 2x

Ex 6.5,8 At what points in the interval [0, 2π ], does the function sin 2𝑥 attain its maximum value? Let f﷐𝑥﷯=﷐sin﷮2𝑥﷯, 𝑥 ∈ ﷐0 , 2𝜋﷯ Step 1: Finding f’﷐𝑥﷯ f’﷐𝑥﷯=﷐𝑑﷐﷐sin﷮2𝑥﷯﷯﷮𝑑𝑥﷯ f’﷐𝑥﷯=2﷐cos﷮2𝑥﷯ Step 2: Putting f’﷐𝑥﷯=0 2cos 2𝑥=0 cos 2𝑥=0 cos 2𝑥=﷐cos﷮﷐𝜋﷮2﷯﷯ 2𝑥=2𝑛𝜋+﷐𝜋﷮2﷯ 𝑥=﷐2𝑛+1﷯﷐𝜋﷮4﷯ Putting n = 0 𝑥=﷐0+1﷯﷐𝜋﷮4﷯= ﷐𝜋﷮4﷯ Putting n = 1 𝑥=﷐2+1﷯﷐𝜋﷮4﷯= ﷐3𝜋﷮4﷯ Putting n = 2 𝑥=﷐2﷐2﷯+1﷯﷐𝜋﷮4﷯= ﷐5𝜋﷮4﷯ Putting n = 3 𝑥=﷐2﷐3﷯+1﷯﷐𝜋﷮4﷯= ﷐7𝜋﷮4﷯ Putting n = 4 𝑥=﷐2﷐4﷯+1﷯﷐𝜋﷮4﷯= ﷐9𝜋﷮4﷯ Putting n = 5 𝑥=﷐2﷐5﷯+1﷯﷐𝜋﷮4﷯ =﷐11𝜋﷮4﷯>2𝜋 Since 𝑥 ∈ ﷐0 , 2𝜋﷯ So, Critical Point are 𝑥 = 0 , ﷐𝜋﷮4﷯ , ﷐3𝜋﷮4﷯ , ﷐5𝜋﷮4﷯ , ﷐7𝜋﷮4﷯ & 2π Step 3: Finding value of f(x) at critical points i.e. 𝑥 = 0 , ﷐𝜋﷮4﷯ , ﷐3𝜋﷮4﷯ , ﷐5𝜋﷮4﷯ , ﷐7𝜋﷮4﷯ & 2π