Let $f: \mathbb{R} \to \mathbb{R}$ be defined as
$$f(x) = \begin{cases} x^5 \sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5 \cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$
The value of $\lambda$ for which $f''(0)$ exists is ____.
(A) 0
(B) 1
(C) $-1$
(D) $\frac{1}{2}$