# Derivatives near optimal points

On page 52 of What Sustains Social Norms and How They Evolve?, Assumption 1 is as follows:

Assumption 1: $$d$$ is continuously differentiable, $$d'(x) \ge 0$$ for all $$x<0$$ and $$d'(x) \leq 0$$ for all $$x>0$$ (it follows that $$d'(0)=0$$).

I don't understand the use of $$\ge$$ and $$\le$$ here. If $$d'(0)=0$$ implies a maximum is obtained at $$x=0$$, then every $$x$$ to the left of $$0$$ should have a strictly positive derivative and every $$x$$ to the right of $$0$$ should have a strictly negative derivative (i.e., $$>$$ and $$<$$ should be used).

What is the meaning behind $$d'(x) \ge 0$$ and $$d'(x) \le 0$$?

Edit: $$d(g-\eta)$$ is a function representing the disutility from social disapproval, where $$g$$ is a tip at a restaurant as a percentage of the bill and $$\eta$$ is the norm tip. The consumer takes as given $$\eta$$ (i.e., the social norm) and chooses $$g$$ to minimise the social disutility $$d(x)$$; i.e, bring it closer to $$0$$, where $$x=g-\eta$$.

Note: there is a trade-off to minimising $$d$$ that is probably not necessary to understand for my question.

• The question would be somewhat more self contained if you were to include what $d$ is meant to describe. Aug 3, 2022 at 6:46
• @Giskard I have included a brief description of $d$. Aug 3, 2022 at 6:57

Assumption 1 ensures that $$x=0$$ is a maximum, but perhaps not the only maximum. It is possible that function does not have a 'peak' at 0, but a plateau. E.g.;
$$f(x) = \left\{ \begin{array}{cl} -(x+1)^2 & \text { if } \ x < -1 \\ 0 & \text { if } \ - 1 \leq x \leq 1 \\ -(x-1)^2 & \text { if } \ 1 < x \end{array} \right.$$ 