What is the maximum value of the function logx-x?

Options

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

Explanation

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

### Step 1: Find the first derivative
We start by differentiating the function \( f(x) \).

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

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

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

Solving for \( x \):

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

### Step 3: Find the second derivative
Next, we find the second derivative to determine the nature of the critical point.

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

### Step 4: 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 maximum at \( x = 1 \).

### Step 5: Find the maximum value
Finally, we substitute \( x = 1 \) back into the original function to find the maximum value.

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

Thus, the maximum value of the function \( f(x) = \log x - x \) is \( -1 \).

### Final Answer
The maximum value of the function \( \log x - x \) is \(-1\).

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