interesting question about mean value theorem

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I have following doubt.

Mean value theorem states for any continuous differential function $f$ over $(a,b)$,

There exists some $c$ such that $f(b)-f(a)=f'(c)(b-a)$

My question is for any $d$, can we find an $i$ and $j$ such that

$f'(d)<=\frac{(f(i)-f(j))}{(i-j)}?$ Can we extend the argument to multi-variable function? Thanks