EDB — 1DD

Exercises

1. [1DD]Let \(I\) be an open interval and \(x_ 0∈ I\), let \(f:I→ℝ\) be differentiable in \(I\) and such that there exists the second derivative \(f''\) in \(x_ 0\): then show that the limit exists

   \[ \lim _{t→ 0}\frac{f(x_ 0+t)+f(x_ 0-t)-2f(x_ 0)}{t^ 2} \]

   and that it coincides with \(f''(x_ 0)\).

   Find then a simple example of \(f\) differentiable in \((-1,1)\) and such that the second derivative \(f''\) in \(x_ 0=0\) does not exist, but the previous limit exists.

   Solution 1

   [1DF]↺↻