JEE Advanced2018MathematicsApplication of DerivativesActual
For every twice differentiable function f : R → - 2 , 2 w i t h f 0 2 + f ′ 0 2 = 85 , which of the following statement(s) is (are) TRUE?
Options
- AThere exist r , s ∈ R , w h e r e r < s , such that f is one-one on the open interval ( r , s )
- BThere exists x 0 ∈ - 4,0 such that f ′ x 0 ≤ 1
- Clim x → ∞ f x = 1
- DThere exists α ∈ - 4 , 4 such that f α + f ′ ′ α = 0 a n d f ′ α ≠ 0
Correct answer
A. There exist r , s ∈ R , w h e r e r < s , such that f is one-one on the open interval ( r , s )
Step-by-step solution
f x can't be constant throughout the domain. Hence we can find x ∈ r , s such that ƒ ( x ) is one-one option ( A ) is true. ( B ) f ′ x = f 0 - f - 4 4 ≤ 1 L M V T ( C ) f x = sin 85 x satisfies given condition But lim x → ∞ sin 85 D . N . E ⇒ Incorrect D g x = f 2 x + f ′ x 2 f ′ x 1 ≤ 1 (by LMVT) f x 1 ≤ 2 (given) ⇒ g x 1 ≤ 5 ∃ x 1 ∈ - 4,0 Similarly g x 2 ≤ 5 ∃ x 2 ∈ 0,4 g 0 = 85 ⇒ g x has maxima in x 1 , x 2 s a y a t α g ′ α = 0 a n d g α ≥ 85 2 f ′ α f α + f ′ ′