$\epsilon$-$\delta$ proof involving differentiation in a defined neighborhood The problem states: Suppose $f'(b) = M$ and $M <0$. Find $\delta>0$ so that if $x\in (b-\delta, b)$, then $f(x) > f(b).$
This intuitively makes sense, but I am not exactly sure how to find $\delta$. I greatly appreciate any help I can receive.  

In general you won't be able to 'find $\delta$' explicitly with only information about $f^\prime$, you can only guarantee its existence based on the the limit definition of the derivative.  This is because the value of $\delta$ will depend on the curvature of the function at $b$, i.e. the second derivative (which, unless explicitly stated, doesn't necessarily exist).
