Question
Question: If the expression \(\left( mx - 1 + \frac{1}{x} \right)\) is non-negative for all positive real x, t...
If the expression (mx−1+x1) is non-negative for all positive real x, then the minimum value of m must be
A
–1/2
B
0
C
¼
D
½
Answer
¼
Explanation
Solution
We know that ax2 + bx + c ≥ 0 if a > 0 and b2 – 4ac ≤ 0.
So, mx – 1 + x1 ≥ 0 ⇒ xmx2−x+1≥0
⇒ mx2 – x + 1 ≥ 0 as x > 0.
Now, mx2 – x + 1 ≥ 0 if m > 0 and 1 – 4m ≤ 0
or if m > 0 and m ≥ 1/4.
Thus, the minimum value of m is ¼