Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12     1. Chapter 6 Class 12 Application of Derivatives
2. Serial order wise
3. Ex 6.2

Transcript

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 0 < x < 1.57 099 < x99 < (1.57)99 0 < x99 < 2.47 × 1099 0 < 100x99 < 247 × 1019 0 < x < 1 ∴ 0 < x < 𝜋/2 So x is in 1st quadrant ∴ cos x is positive (As 𝜋/2 = 3.14/2 = 1.57 > 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 1.57 < x < 3.14 (1.57)99 < x99 < (3.14)99 2.47 × 1099 < x99 < 3.14 × 1099 247 × 1099 < 100 x99 < 314 × 1099 Since 𝜋/2 < x < 𝜋 so x is in 2nd quadrant ∴ cos x is negative. Minimum value of cos x = −1 (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) 0<𝑥< 𝜋/2 0 < x < 1.57 099 < x99 < (1.57)99 0 < x99 < 2.47 × 1099 0 < cos x < 𝜋/2 So x is in 1st quadrant ∴ cos x is positive Thus, f(x) is strictly decreasing for none of the intervals. So, (D) is the correct answer

Ex 6.2 