Suppose a differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ has a local maximum at $(a, f(a))$ (This means there are numbers $m$ and $M$ such that (i) $m < a < M$ and (ii) $f(a) \geq f(x)$ for any $x \in [m,M]$.) The proof of a standard result is sketched below. Complete it as instructed.
Proof: For sufficiently $\_\_\_\_$ 1 $h > 0$, it is given that $f(a+h)$ $\_\_\_\_$ 2 3. Therefore for such $h$ the quantity $\_\_\_\_$ 4 must be $\_\_\_\_$ 5. By taking the limit of this quantity as $h \rightarrow 0$ from the right, we get that $\_\_\_\_$ 7 must be $\_\_\_\_$ 8. A parallel argument for suitable negative values of $h$ gives that $\_\_\_\_$ 10 must be $\_\_\_\_$ 11. Combining both conclusions gives the desired result: $\_\_\_\_$ 13 $\_\_\_\_$ 14. Note that the mentioned limits exist because $\_\_\_\_$ 16.
Write a sequence of 9 letters indicating the correct options to fill in the numbered blanks 1 to 9. Do not use any spaces, full stop or other punctuation. E.g., ABACDIJKB is in the correct format. [3 points]
Options: A. small B. large C. $\geq$ D. $>$ E. $\leq$ F. $<$ G. $=$ H. $\neq$ I. 0 J. $f(a)$ K. $\frac{f(a+h)-f(a)}{h}$ L. $f'(a)$ M. $f$ is differentiable N. $f$ is continuous