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………

Given f'(x)<=2

Integrate both side, we'll get f(x)=2x+C

From Given , f(1)=2(1)+C=2 

We Get C=0;

Also F(4)=2(4)+0 = 8 (Given)

Thus F(2)=2(2)=4

Please Reply if i am Wrong.

Using LMVT for f in [1, 2]
2 1
(2) (1)
−
f − f = f ′(c) ≤ 2 ⇒ f (2) – f (1) ≤ 2
⇒ f(2) ≤ 4
Using LMVT for f in [2, 4]
4 2
(4) (2)
−
f − f = f ′(d) ≤ 2 and proceed