Let f : R → R be defined as f x = x 5 sin 1 x + 5 x 2 , x < 0 0 , x = 0 x 5 cos 1 x + λ x 2 , x > 0 . The value of λ for which f " ( 0 ) exists, is___.

Step-by-step solution

f x = 5 x 4 · sin ( 1 / x ) - x 3 cos ( 1 / x ) + 10 x , x < 0 0 , x = 0 5 x 4 · cos ( 1 / x ) + x 3 sin ( 1 / x ) + 2 λ x , x > 0 Differentiating two times. f " x = 20 x 3 · sin ( 1 / x ) - 5 x 2 cos ( 1 / x ) - 3 x 2 cos ( 1 / x ) - x sin ( 1 / x ) + 10 ,   x < 0 20 x 3 · cos ( 1 / x ) + 5 x 2 sin ( 1 / x ) + 3 x 2 sin ( 1 / x ) - x cos ( 1 / x ) + 2 λ ,   x > 0 f " 0 + = f " 0 - ⇒ 2 λ = 10   ⇒   λ = 5