Ex 6.2, 13 On which intervals f (x) = x100 + sin ⁡x -1 strictly - Ex 6.2

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  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.2,13 On which of the following intervals is the function f given by f (𝑥) = ﷐𝑥﷮100﷯ + sin⁡𝑥 –1 strictly decreasing ? (A) (0,1) (B) ﷐﷐𝜋﷮2﷯,𝜋﷯ (C) ﷐0,﷐𝜋﷮2﷯﷯ (D) None of these f (x) = 𝑥100 + sin x − 1 f’(x) = 100𝑥99 + cos x (A) (0, 1) For 0 < x < 1 So, 100 x99 + cos x is positive. ∴ f’(x) > 0 f (x) is increasing on (0, 1) (B) ﷐﷐𝝅﷮𝟐﷯, 𝝅﷯ ﷐𝜋﷮2﷯ < x < 𝜋 1.57 < x < 3.14 Since 100 x99 is much greater than –1 So 100 x99 + cos x is positive ∴ f’ (x) > 0 ﷐𝑪﷯﷐𝟎, ﷐𝝅﷮𝟐﷯﷯ 0<𝑥< ﷐𝜋﷮2﷯ So 100 x99 + cos x is positive ∴ f’ (x) > 0 f(x) is increasing on ﷐0,﷐𝜋﷮2﷯﷯ f(x) is strictly decreasing for none of the intervals. So (D) is the correct answer

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