f(x) = x, g(x) = |x|
Also
\((h(x) - f(x))^2 + (h(x - g(x)))^4 \le 0\)
⇒ \(h(x) - f(x) = 0 \) and \(h(x) - g(x) = 0\)
(∵ h(x) is real valued function & if \(a^2+ b^2 \le 0 \) then a = 0 & b = 0)
⇒ \(h(x) = f(x) = x\; and\; h(x) = g(x) = |x|\)
⇒ \(x = |x|\) which is true only when \(x \ge 0\)
Hence, \(h(x) = x,\text{if} x\in [0, \infty).\)
∴ Option (C) is correct.
Also
\(h(5) = 5, f(5) = 5 \) & \(g(5) = |5| = 5\)
∴ \(2h(5) + f(5) + g(5) = 10 + 5 + 5= 20\)
Hence, option (A) is also correct.