Let f be any function continuous on [a, b] and twice differentiable on (a, b).

Question

Let f be any function continuous on [a, b] and
twice differentiable on (a, b). If for all x $$ \in $$ (a, b),
f'(x) > 0 and f''(x) < 0, then for any c $$ \in $$ (a, b),
$${{f(c) - f(a)} \over {f(b) - f(c)}}$$ is greater than :

Accepted Answer

It is clear from graph that, slope of AC $>$ slope of CB

$$ \Rightarrow $$ $${{f\left( c \right) - f\left( a \right)} \over {c - a}}$$ $>$ $${{f\left( b \right) - f\left( c \right)} \over {b - c}}$$

$$ \Rightarrow $$ $${{f\left( c \right) - f\left( a \right)} \over {f\left( b \right) - f\left( c \right)}}$$ $$ > {{c - a} \over {b - c}}$$

Solutions · 1