Given some function $f: I \subseteq\mathbb R \rightarrow \mathbb R$, Which is differentiable twice at some point $a\in I$.
Prove:
If $f''(a)>0$ then, $f(a)+f'(a)(x-a)\leq f(x)$ in a region of $a$.
I think we can't use the Lagrange remainder for taylor polynomials,
because we can't assume that the function is differentiable around $a$.
Best Answer
$f''(a) >0$ implies that $f'(x) >f'(a)$ for $a <x <a+\delta$ for some $\delta >0$. [This follows from definition of $f''(a)$]. Now $a <x <a+\delta$ implies $f(x)-f(a)=(x-a)f'(t)$ for some $t \in (a,x)$ so $f(x)-f(a)\geq (x-a)f'(a)$. Similar argument works to the left of $a$.