Question

Let a, b, c be three real numbers such that a < b < c. Let f(x) be continuous $\forall x\in [a,c]$ and differentiable $\forall x\in (a,c)$. If $f^{\prime \prime }\left(x\right)>0\forall x\in (a,c)$ then

• A. $(c-b)f\left(a\right)+(b-a)f\left(c\right)>(c-a)f\left(b\right)$
• B. $(c-b)f\left(a\right)+(a-c)f\left(b\right)<(a-b)f\left(c\right)$
• C. $f\left(a\right)<f\left(b\right)<f\left(c\right)$
• D. None of these

Answer 1

The correct option is A

$(c-b)f\left(a\right)+(b-a)f\left(c\right)>(c-a)f\left(b\right)$

By LMVT,

$\frac{f\left(b\right)-f\left(a\right)}{b-a}<\frac{f\left(c\right)-f\left(b\right)}{c-b}$