Question

If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n"  "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.

Options

• 2.00

• 3.00

• 4.00

• 5.00

Answer 1

If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `"d"/("d"x) underbrace(((fofof......of)(x)))_("n"  "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is 4.00.

Explanation:

`f(x) = (x + 1)/4`

⇒ `(fof)(x) = (x + 1 + 4)/4^2`

⇒ `(fofof)(x) = (x + 1 + 4 + 4^2)/4^3`

∴ `underbrace((fofof....f(x)))_("n"  "times") = (x + 1 + 4 + 4^2 + .... + 4^("n" - 1))/4^"n"`

= `(3x + 4^"n" - 1)/3.4^"n"`

`"d"/("d"x)(fofo .......f(x))""|_(x = 1/2) = 1/4^"n"`

⇒ m = 4