Let y = 3 + sin x, y ∈ [2, 4] and assumes all values there in.

Also, let g(y) = y + , this function is increasing on [2, 4].

So, g(2) ≤ g(y) ≤ g(4).

Thus, 3 ≤ g(y) ≤ and both external values are attained.

It now follows that the minimum of

f(b) = max(|g(y) + b – 3|) is ,

which is attained at b =

For if b > , then choose x = , so y = 4 and then

g(y) + b – 3 >

While if b < , then choose x = , so y = 2 and

g(y) + b – 3 =

On the other hand range for g(y) is

≤ g(y) + b – 3 ≤ for b = .

∴ Minimum value of f(b) is .