Suppose function $f\left(x\right)$ is defined on a closed interval $\left[a, b\right]$. Is "$f\left(x\right)$ is twice differentiable on $\left(a,b\right)$ and $f''\left(x\right)>0$ on $\left(a, b\right)$" sufficient for $f$ being strictly convex on $\left[a,b\right]$? (I want $f\left(\lambda a+\left(1-\lambda\right)b\right)<\lambda f\left(a\right)+\left(1-\lambda\right)f\left(b\right)$, for all $\lambda \in \left(0,1\right)$) (I think the answer is no because I can arbitrarily change the function value at $a$ and $b$ without violating these conditions.) Do we need the additional condition that "the second order right derivative exists at $a$ and is strictly positive, and the second order left derivative exists at $b$ is strictly positive"? Is it possible to make the additional condition weaker?
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