*(This question, "Let f: R R...", is from Continuity and Differentiability, not Straight L…

Question

*(This question, "Let $f:\mathbb R	o\mathbb R$...", is from Continuity and Differentiability, not Straight Lines --- appears misplaced in the source PDF. Reproduced here for completeness as it was numbered in the source.)*

Let $f(x) = \begin{cases}x^5\sin\frac1x+5x^2 & x<0\0 & x=0\x^5\cos\frac1x+\lambda x^2 & x>0\end{cases}$. The value of $\lambda$ for which $f''(0)$ exists, is \_\_\_\_\_\_.

Answer: 5

Accepted Answer

$f''(0^+) = f''(0^-)$ requires $2\lambda = 10 \implies \lambda=5$ (from matching the constant terms in the second derivatives on each side, since the oscillatory terms vanish as $x	o0$).

Solutions · 1