Lot of be any function continuous on [a, b ] and twice differentiable on (a, b). If for all x ϵ (a, b ) , f'( x ) 0 and f'' ( x )

Question

Lot of be any function continuous on [a, b ] and twice differentiable on (a, b). If for all x ϵ (a, b ) , f'( x ) > 0 and f'' ( x ) < 0, then for any c ϵ (a, b), (f(c) - f(a)/f(b)-f(c)) is greater than : (A) (b-c/c-a) (B) 1 (C) (c-a/b-c) (D) (b+a/b-a)

Tardigrade Admin · Accepted Answer

it is clear from graph that m1 > m2 ⇒ (f (c)-f (a)/c-a) > (f (b)-f (c)/b-c) ( f (c)-f (a)/f (b)-f (c)) > (c-a/b-c)

Q. Lot of be any function continuous on [ $a, b$ ] and twice differentiable on (a, b). If for all x $ϵ$ ( $a, b$ ) , $f^{'} (x) > 0 an d f^{''} (x) < 0$ , then for any $cϵ$ ( $a, b$ ), $\frac{f ( c ) - f ( a )}{f ( b ) - f ( c )}$ is greater than :

$\frac{b - c}{c - a}$

$1$

$\frac{c - a}{b - c}$

$\frac{b + a}{b - a}$

it is clear from graph that $m_{1} > m_{2}$
$\Rightarrow \frac{f ( c ) - f ( a )}{c - a} > \frac{f ( b ) - f ( c )}{b - c}$
$\frac{f ( c ) - f ( a )}{f ( b ) - f ( c )} > \frac{c - a}{b - c}$

Q. Lot of be any function continuous on [a,b ] and twice differentiable on (a, b). If for all x ϵ (a,b ) , f′(x)>0andf′′(x)<0, then for any cϵ (a,b), f(b)−f(c)f(c)−f(a)​ is greater than :

c−ab−c​

1

b−cc−a​

b−ab+a​

it is clear from graph that m1​>m2​ ⇒c−af(c)−f(a)​>b−cf(b)−f(c)​ f(b)−f(c)f(c)−f(a)​>b−cc−a​

Q. Lot of be any function continuous on [ $a, b$ ] and twice differentiable on (a, b). If for all x $ϵ$ ( $a, b$ ) , $f^{'} (x) > 0 an d f^{''} (x) < 0$ , then for any $cϵ$ ( $a, b$ ), $\frac{f ( c ) - f ( a )}{f ( b ) - f ( c )}$ is greater than :

$\frac{b - c}{c - a}$

$1$

$\frac{c - a}{b - c}$

$\frac{b + a}{b - a}$

it is clear from graph that $m_{1} > m_{2}$
$\Rightarrow \frac{f ( c ) - f ( a )}{c - a} > \frac{f ( b ) - f ( c )}{b - c}$
$\frac{f ( c ) - f ( a )}{f ( b ) - f ( c )} > \frac{c - a}{b - c}$