Question: Given \( x > 0 \), find the maximum of the expression \( \frac{4}{x + 1} - \frac{1}{x} \).

Given \( x > 0 \), find the maximum of the expression \( \frac{4}{x + 1} - \frac{1}{x} \).

Solution

Step 1: Let us denote the expression as \( f(x) = \frac{4}{x + 1} - \frac{1}{x} \). Step 2: To find the maximum, we take the derivative of \( f(x) \) with respect to \( x \). \[ f’(x) = -\frac{4}{(x + 1)^2} + \frac{1}{x^2} \] Step 3: Set the derivative equal to zero and solve for \( x \). \[ -\frac{4}{(x + 1)^2} + \frac{1}{x^2} = 0 \] Step 4: Rearrange the equation to find a common denominator. \[ \frac{1}{x^2} = \frac{4}{(x + 1)^2} \] Step 5: Take the square root of both sides. \[ \frac{1}{x} = \frac{2}{x + 1} \] Step 6: Solve for \( x \). \[ 2x = x + 1 \\ x = 1 \] Step 7: Verify that this critical point gives a maximum by evaluating the second derivative or testing the behavior of \( f(x) \) around \( x = 1 \). Step 8: Substitute \( x = 1 \) into the original expression to find the maximum value. \[ f(1) = \frac{4}{1 + 1} - \frac{1}{1} = 2 - 1 = 1 \] The maximum value of the expression is \( 1 \).