JEE Advanced2015MathematicsApplication of DerivativesActual
Paragraph: Let F: R R be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F^ (x) Question: The correct statement(s) is(are)
Options
- Af ′ 1 < 0
- Bf 2 < 0
- Cf ′ x ≠ 0 f o r a n y x ∈ ( 1 , 3 )
- Df ′ x = 0 f o r s o m e x ∈ ( 1 , 3 )
Correct answer
A. f ′ 1 < 0
Step-by-step solution
F 1 = 0 , F 3 = - 4 F ′ x < 0 F ( x ) is decreasing, also F x < 0 Now, f x = x F ( x ) x ∈ 1 2 , 3 x F x < 0 ∴ f 2 < 0 Also, f ′ x = x F ′ x + F ( x ) f ′ 1 = F ′ 1 < 0 Also, f ( x ) is strictly decreasing Now, F 1 = F x = 0 ∴ f ′ x ≠ 0 a s F ′ x < 0 a n d f x is decreasing.