comparison of derivatives

Question

I have a basic question related to inequalities:

Suppose that $A(x) \leq B(x) $ in $[a, b]$.

Then under what hypotheses $$ \frac{d}{dx} A(x) \leq \frac{d}{dx} B(x) $$ in $[a, b]$?

So far I am thinking about monotonicity of both A and B in $[a,b]$. I would like to know all of the implicit hypotheses if I use the statement in a proof.

Answer 1

not always true. for example, consider $A = x $ and $B= x^2$. We know $A(x) \geq B(x)$ for all $x \in [0,1]$, but, $A' = 1 $ and $B' = 2x $ and

$$ 1 \geq 2x \; \; \; \text{does not hold for all} \; \; x \in [0,1] $$