Let f(x)=x^3. 1. f has a local maximum at x=0. 2. f'(x) changes sign at x=0.

Here $f'(x)=3x^2\ge0$ for all $x$, so $f$ is increasing everywhere. Hence there is no local maximum at 0, and statement 1 is false. Since $f'$ is non-negative on both sides of 0, it does not change sign, so statement 2 is false. Trap: $f'(0)=0$ suggests an extremum, but a stationary point need not be one; $x=0$ is a point of inflection.