Let f(x) be a polynomial satisfies the relation f(f(f(x)) + (1 – p) f(x) = 3, \forall x \in R, where p is any real number. If leading coefficient of f(x) is 2 then find the value of \left(\right. \frac{d}{dx} \left(\right. f \left(\right. f \left(\right. x \left.\right) \left.\right) \left.\right) \textrm{ }\textrm{ } at \textrm{ }\textrm{ } x = p \left.\right) + \int_{- p}^{p} \left(\right. 2 f^{- 1} \left(\right. x \left.\right) + 1 \left.\right) \textrm{ } dx .

Engineering

Mathematics

Properties of Definite Integral

Question

Let f(x) be a polynomial satisfies the relation f(f(f(x)) + (1 – p) f(x) = 3, $\forall$ x $\in$ R, where p is any real number. If leading coefficient of f(x) is 2 then find the value of $(\frac{d}{dx} (f (f (x))) at x = p) + \int_{- p}^{p} (2 f^{- 1} (x) + 1) dx .$

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Let degree of f(x) be n.

$∵$ degree of f(f(f(x))) = degree of f(x)

⇒ n³ = n ⇒ n = 1

$∴$ Let f(x) = 2x + b

( $∵$ leading coefficient = 2)

$∴$ f(f(f(x))) = 2(4x + 3b) + b = 8x + 7b

$∴$ 8 + 2 – 2p = 0 ⇒ p = 5

and 7b – 4b = 3 ⇒ b = 1

$∴$ f (x) = 2x + 1

f(f(x)) = 4x + 3

$∴$ $\frac{d}{dx} f (f (x)) = 4$

f^–1(x) = $\frac{x - 1}{2}$ ⇒ 2f^–1(x) + 1 = x

$∴$ $\int_{- p}^{p} (2 f^{- 1} (x) + 1) dx = 0$