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  4. Suppose, that f is differentiable for all x and that f'(x) ≤ 2 for all x. If f(1) = 2 and f(4) = 8, then f(2) has the value equal to

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Q. Suppose, that $f$ is differentiable for all $x$ and that $f'(x) \leq 2$ for all x. If $f(1) = 2$ and $f(4) = 8$, then $f(2)$ has the value equal to

Application of Derivatives

Solution:

$LMV$ Theorem for $f$ in $[1, 2]$
$\forall\,c \in\left(1, 2\right) \frac{f\left(2\right)-f\left(1\right)}{2-1}$
$=f'\left(c\right) \le 2$
$f\left(2\right)-f\left(1\right) \le 2$
$\Rightarrow f\left(2\right) \le 4 \quad...(1)$
Again, using $LMV$ Theorem in $\left[2, 4\right]$
$\forall\,d \in\left(2, 4\right) \frac{f\left(4\right)-f\left(2\right)}{4-2}$
$=f'\left(d\right) \le 2$
$\therefore f\left(4\right)-f\left(2\right) \le 4$
$\Rightarrow 8-f\left(2\right) \le 4$
$\Rightarrow 4 \le f\left(2\right)$
$\Rightarrow f\left(2\right) \ge 4$
From $\left(1\right)$ and $\left(2\right)$, we get $f\left(2\right)=4$$LMV$ Theorem for $f$ in $[1, 2]$
$\forall\,c \in\left(1, 2\right) \frac{f\left(2\right)-f\left(1\right)}{2-1}$
$=f'\left(c\right) \le 2$
$f\left(2\right)-f\left(1\right) \le 2$
$\Rightarrow f\left(2\right) \le 4 \quad...(1)$
Again, using $LMV$ Theorem in $\left[2, 4\right]$
$\forall\,d \in\left(2, 4\right) \frac{f\left(4\right)-f\left(2\right)}{4-2}$
$=f'\left(d\right) \le 2$
$\therefore f\left(4\right)-f\left(2\right) \le 4$
$\Rightarrow 8-f\left(2\right) \le 4$
$\Rightarrow 4 \le f\left(2\right)$
$\Rightarrow f\left(2\right) \ge 4 \quad ...(2)$
From $\left(1\right)$ and $\left(2\right)$, we get $f\left(2\right)=4$