Q.14 The minimum value of the function
𝑓(𝑥) = 𝑥 +4/𝑥
for 𝑥 > 0 is
(A) 1
(B) 2
(C) 3
(D) 4

The function f(x)=x+4/x for x>0 achieves its minimum value of 4 at x=2, confirmed through differentiation and inequality methods.

Derivative Method

To find the minimum, compute the first derivative: f′(x)=1−4/x². Set f′(x)=0, yielding x²=4, so x=2 (since x>0). The second derivative f′′(x)=8/x³>0 at x=2, confirming a local minimum. Substituting gives f(2)=2+2=4.

AM-GM Inequality Proof

By AM-GM inequality, x + 4/x ≥ 2√(x·4/x) = 2√4 = 4, with equality when x = 4/x, or x=2. This elegantly shows the minimum is exactly 4 for x>0.

Options Analysis

Option	Value	Explanation
(A)	1	Incorrect: f(x)≥4>1; no x yields 1, as f(1)=5.
(B)	2	Incorrect: Minimum exceeds 2; e.g., f(4)=5, f(0.5)=10.
(C)	3	Incorrect: AM-GM bound is 4, not 3; function values always ≥4.
(D)	4	Correct: Achieved at x=2 via calculus or AM-GM equality case.