Misc 14 - Find absolute max, min values f(x) = cos2 x + sin x - Absolute minima/maxima

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  1. Chapter 6 Class 12 Application of Derivatives
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Misc 14 Find the absolute maximum and minimum values of the function f given by f (𝑥) = cos2 𝑥 + sin⁡𝑥, 𝑥 ∈ [0, 𝜋 ] f﷐𝑥﷯=﷐cos﷮2﷯𝑥+sin 𝑥 , 𝑥 ∈ ﷐0 , 𝜋﷯ Step 1: Finding f’﷐𝑥﷯ f﷐𝑥﷯=﷐cos﷮2﷯𝑥+sin 𝑥 f’﷐𝑥﷯= ﷐𝑑﷐﷐﷐cos﷮2﷯﷮𝑥 + ﷐sin﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ = 2cos 𝑥. ﷐𝑑﷐cos 𝑥﷯﷮𝑑𝑥﷯ + cos 𝑥 = 2cos 𝑥﷐−sin 𝑥﷯+﷐cos﷮𝑥﷯ = cos 𝑥 ﷐−2sin 𝑥+1﷯ Step 2: Putting f’﷐𝑥﷯ = 0 cos 𝑥 ﷐−2﷐sin﷮𝑥+1﷯﷯=0 𝑥 = ﷐𝜋﷮6﷯ & ﷐𝜋 ﷮2﷯ are Critical points. Since our interval is 𝑥 ∈ [0, 𝜋 ] Critical points are 𝑥=0, ﷐𝜋﷮6﷯ , ﷐𝜋﷮2﷯ , 𝜋 Step 3: Hence Absolute maximum value = ﷐𝟓﷮𝟒﷯ & Absolute minimum value = 1

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