What is the maximum value of ihe function log x -x?

Options

A. A. `-1`
B. B. 0
C. C. 1
D. D. `infty`

Answer

To find the maximum value of the function \( f(x) = \log x - x \), we will follow these steps:

### Step 1: Find the first derivative of the function
The first step is to differentiate the function with respect to \( x \).

\[ f'(x) = \frac{d}{dx}(\log x - x) = \frac{1}{x} - 1 \]

### Step 2: Set the first derivative to zero
To find the critical points, we set the first derivative equal to zero:

\[ \frac{1}{x} - 1 = 0 \]

### Step 3: Solve for \( x \)
Now, we solve the equation for \( x \):

\[ \frac{1}{x} = 1 \implies x = 1 \]

### Step 4: Find the second derivative
Next, we need to find the second derivative to determine if this critical point is a maximum or minimum:

\[ f''(x) = \frac{d}{dx}\left(\frac{1}{x} - 1\right) = -\frac{1}{x^2} \]

### Step 5: Evaluate the second derivative at the critical point
Now, we evaluate the second derivative at \( x = 1 \):

\[ f''(1) = -\frac{1}{1^2} = -1 \]

Since \( f''(1) < 0 \), this indicates that the function has a local maximum at \( x = 1 \).

### Step 6: Find the maximum value
Finally, we find the maximum value of the function by substituting \( x = 1 \) back into the original function:

\[ f(1) = \log(1) - 1 = 0 - 1 = -1 \]

Thus, the maximum value of the function \( f(x) = \log x - x \) is:

\[ \boxed{-1} \]