Answer the following True-False quiz. (Enter "T" or "F".)

1. If $f'(c)=0$ and $f''(c)>0$, then $f(x)$ has a local minimum at $c$.
2. If a function has a local maximum at $c$, then $f'(c)$ exists and is equal to 0.
3. If $f'(x)<0$ for all $x$ in $(0,1)$, then $f(x)$ is decreasing on $(0,1)$.
4. A continuous function on a closed interval always attains a maximum and a minimum value.
5. $(f(x)+g(x))' = f'(x) + g'(x)$.
6. Differentiable functions are always continuous.
7. If $f(x)$ and $g(x)$ are increasing on an interval $I$, then $f(x)g(x)$ is increasing on $I$.