Let f : R → R be given by f x = x 5 + 5 x 4 + 10 x 3 + 10 x 2 + 3 x + 1 , x < 0 ; x 2 - x + 1 , 0 ≤ x < 1 ; 2 3 x 3 - 4 x 2 + 7 x - 8 3 , 1 ≤ x < 3 ; x - 2 l o g e x - 2 - x + 10 3 , x ≥ 3 Then which of the following options is/are correct?
D. f ′ has a local maximum at x = 1
As given f : R → R f x = x 5 + 5 x 4 + 10 x 3 + 10 x 2 + 3 x + 1 x 0 x 2 - x + 1 0 ≤ x 1 2 3 x 3 - 4 x 2 + 7 x - 8 3 1 ≤ x 3 ( x - 2 ) l n ( x - 2 ) - x + 10 3 x ≥ 3 f ( 0 ) = f ( 0 + ) = f ( 0 ) = 1 f 1 = f 1 + = f 1 = 1 f 3 = f 3 + = f 3 = 1 3 Hence f x is continuous everywhere. Now, l i m x → ∞ f x = - ∞ and l i m x → ∞ f x = + ∞ Hence, range of f ( x ) is - ∞ , ∞ Consider f ' x = 5 x 4 + 20 x 3 + 30 x 2 + 20 x + 3 = 5 ( x + 1 ) 4 - 2 x 0 2 x - 1 0 x 1 2 x 2 - 8 x + 7 1 x 3 l n ( x - 2 ) x > 3 Option ( B ) : For x 0 f ′ x = 5 x + 1 4 - 2 , which takes both positive and negative values. Hence, f ( x ) is non-monotonic for x 0 Option ( A ) : f ″ x = 20 ( x + 1 ) 3 x 0 2 0 x 1 4 x - 3 1 x 3 1 x - 2 x > 3 Now, f ” 1 ′ = 4 0 and f ” 1 - = 2 > 0 Given of f ′ ( x ) in x ∈ 0,3 x = 1 is maxima for f ′ x -----Option D and f ′ x is not differential at x = 1 Option ( C ) : l i m x → ∞ f x = l i m x → ∞ x - 2 ln x - 2 - x + 10 3 l i m x → ∞ f x = l i m x → ∞ x - 2 ln x - 2 - 1 + 4 3 = ∞ l i m x → - ∞ f x = - ∞ Hence, range of f ∈ - ∞ , ∞
Related: Mathematics — Application of Derivatives · All PYQ Banks