Question

Let f : R → R be defined as f(x) = {(x⁵ sin(1/x) + 5x², x < 0), (0, x = 0), (x⁵ cos(1/x) + λx², x > 0) of λ for which f''(0) exists, is ___.

Answer 1

LHD of f'(x) at x = 0 is 10 

RHD of f'(x) at x = 0 is 2λ

If f"(0) exists then 

2λ = 10 

λ = 5