Q.14 The minimum value of the function
π(π₯) = π₯ +4/π₯
for π₯ > 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: fβ²(x)=1β4/xΒ². Set fβ²(x)=0, yielding xΒ²=4, so x=2 (since x>0). The second derivative fβ²β²(x)=8/xΒ³>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 β₯ 2β(xΒ·4/x) = 2β4 = 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)β₯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 β₯4. |
| (D) | 4 | Correct: Achieved at x=2 via calculus or AM-GM equality case. |
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