Introduction:
To find the minimum value of m in the expression mx-1 1/x, we need to determine the conditions under which the expression is non-negative for all positive real values of x.

Understanding the expression:
Let's break down the expression: mx - 1/x

- The term mx represents a linear function with respect to x, where m is the slope of the line.
- The term 1/x represents a hyperbola, which is always positive for positive values of x.

Conditions for non-negativity:
For the entire expression to be non-negative, both terms mx and 1/x must be non-negative.

Condition 1: mx ≥ 0
- For mx to be non-negative, the slope m must be greater than or equal to zero.
- This condition ensures that the linear function mx does not have negative values.

Condition 2: 1/x ≥ 0
- For 1/x to be non-negative, x must be greater than zero.
- This condition ensures that the hyperbolic term 1/x does not have negative values.

Combining the conditions:
To satisfy both conditions simultaneously, m must be greater than or equal to zero, and x must be greater than zero.

Minimum value of m:
Since the expression mx - 1/x is non-negative for all positive real values of x, the minimum value of m occurs when the expression is equal to zero. This means that mx = 1/x.

- If x approaches zero, 1/x approaches infinity, so mx must also approach infinity for mx - 1/x to be equal to zero.
- Therefore, there is no minimum value of m.

Conclusion:
The expression mx - 1/x is non-negative for all positive real values of x when m is greater than or equal to zero. However, there is no minimum value of m for which the expression is non-negative.

Solving (mx) - 11/x >=0
We get min. value of m as 1/4