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Q.
Suppose that $f$ is differentiable for all $x$ and that $f^{\prime}( x ) \leq 2$ for all $x$. If $f(1)=2$ and $f (4)=8$ then $f(2)$ has the value equal to
Application of Derivatives
Solution:
Using LMVT for $f$ in $[1,2]$
$\forall c \in(1,2) \frac{ f (2)- f (1)}{2-1}= f ^{\prime}( c ) \leq 2 $
$f (2)- f (1) \leq 2 \Rightarrow f (2) \leq 4$....(1)
again using LMVT in $[2,4]$
$\forall d \in(2,4) \frac{ f (4)- f (2)}{4-2}= f ^{\prime}( d ) \leq 2 $
$\therefore f (4)- f (2) \leq 4 $
$8- f (2) \leq 4$
$4 \leq f (2) f (2)=4$....(2)
from (1) and$ (2) f (2) = 4$