Suppose that f is differentiable for all x and that f '(x) le 2 for all x. If f (1) = 2 and f (4

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5. Continuity and Differentiation

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

A

$3$

B

$4$

C

$6$

D

$8$

Solution

Using $LMVT$ for $f$ in $[1, 2]$

$\forall c \in (1, 2)$ $\frac{{f(2) – f(1)}}{{2 – 1}}= f ' (c) \le 2$

$f (2) – f (1) \le 2==>f (2) \le 4$$….(1)$

again using $LMVT$ in $[2, 4]$

$\forall d \in (2, 4)\,\,\frac{{f(4) – f(2)}}{{4 – 2}}= f ' (d) \le 2 $

$f (4) – f (2) \le 4$

$8 – f (2) \le 4$

$4 \le f (2)$==>$f (2) \ge 4$.$…(2)$

from $(1)$ and $(2)$

$ f (2) = 4$

Standard 12

Mathematics

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