Question

Let \( f(x)=x \) and \( g(x)=|x| \) for all \( x \in R \). Then the function \( h(x) \) satisfying \( (h(x)-f(x))^{2}+(h(x)-g(x))^{4} \leq 0 \), then (1) \( 2 h(5)+f(5)+g(5)=20 \) (2) \( h(x)=x \), if \( x \in(-5,15) \) (3) \( h(x)=x \), if \( x \in[0, \infty) \) (4) \( h(x)=x \), if \( x \in R \)

Answer 1

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.