Question

If $f\left(x\right)$ is a differentiable function satisfying $\left|f^{'} \left(x\right)\right|\leq 4\forall x\in \left[0 , 4\right]$ and $f\left(0\right)=0,$ then

• A. $f\left(x\right)=18$ has no solution in $x\in \left[0 , 4\right]$
• B. $f\left(x\right)=18$ has more than $2$ solutions in $x\in \left[0 , 4\right]$
• C. $f\left(x\right)=14$ has no solution in $x\in \left[0 , 4\right]$
• D. $f\left(x\right)=20$ has $2$ solution in $x\in \left[0 , 4\right]$

Answer 1

Answer: (A)

Applying LMVT in $x\in \left[0 , t\right]$ for $f\left(x\right),$ we get
$f^{'} \left(c\right) = \frac{f \left(t\right) - f \left(0\right)}{t - 0}$
$\Rightarrow \left|f^{'} \left(c\right)\right| = \left|\frac{f \left(t\right)}{t}\right| \leq 4$
$\Rightarrow \left|f \left(t\right)\right|\leq 4t$
As $t\in \left[0 , 4\right]$
$\therefore \left|f \left(t\right)\right|\leq 16$