The curve $y=log_{e}x$ is:

Q: The curve $y=log_{e}x$ is:

The correct answer is Option 2: The curve $ y = \log_{e}x $ is concave downwards in $ (0, \infty) $ only. ### Explanation: 1. **Understanding Concavity:** - A function is concave downwards on an interval if its second derivative is negative on that interval. - For the function $ y = \log_{e}x $, we need to determine the sign of its second derivative to understand its concavity. 2. **First Derivative:** - The first derivative of $ y = \log_{e}x $ is $ \frac{d}{dx}(\log_{e}x) = \frac{1}{x} $. 3. **Second Derivative:** - The second derivative is $ \frac{d^2}{dx^2}(\log_{e}x) = \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} $. - Notice that $ -\frac{1}{x^2} $ is negative for all $ x > 0 $. 4. **Conclusion on Concavity:** - Since the second derivative $ -\frac{1}{x^2} $ is negative for all $ x > 0 $, the function $ y = \log_{e}x $ is concave downwards on the interval $ (0, \infty) $. ### Why Other Options Are Incorrect: 1. **Option 1: Concave upwards in $ (0, \infty) $ only** - This is incorrect because the second derivative $ -\frac{1}{x^2} $ is negative, indicating concave downwards, not upwards. 2. **Option 3: Concave upwards in $ (-\infty, \infty) $** - This is incorrect because the domain of $ y = \log_{e}x $ is $ (0, \infty) $, and within this domain, the function is concave downwards, not upwards. 3. **Option 4: Concave downwards in $ (-\infty, \infty) $** - This is incorrect because the function $ y = \log_{e}x $ is only defined for $ x > 0 $. The interval $ (-\infty, \infty) $ includes negative values and zero, where the logarithmic function is not defined. In summary, the function $ y = \log_{e}x $ is concave downwards in its domain $ (0, \infty) $, which supports Option 2 as the correct answer.

Question

Accepted Answer

The correct answer is Option 2: The curve $ y = \log_{e}x $ is concave downwards in $ (0, \infty) $ only.

### Explanation:

1. **Understanding Concavity:**
   - A function is concave downwards on an interval if its second derivative is negative on that interval.
   - For the function $ y = \log_{e}x $, we need to determine the sign of its second derivative to understand its concavity.

2. **First Derivative:**
   - The first derivative of $ y = \log_{e}x $ is $ \frac{d}{dx}(\log_{e}x) = \frac{1}{x} $.

3. **Second Derivative:**
   - The second derivative is $ \frac{d^2}{dx^2}(\log_{e}x) = \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} $.
   - Notice that $ -\frac{1}{x^2} $ is negative for all $ x > 0 $.

4. **Conclusion on Concavity:**
   - Since the second derivative $ -\frac{1}{x^2} $ is negative for all $ x > 0 $, the function $ y = \log_{e}x $ is concave downwards on the interval $ (0, \infty) $.

### Why Other Options Are Incorrect:

1. **Option 1: Concave upwards in $ (0, \infty) $ only**
   - This is incorrect because the second derivative $ -\frac{1}{x^2} $ is negative, indicating concave downwards, not upwards.

2. **Option 3: Concave upwards in $ (-\infty, \infty) $**
   - This is incorrect because the domain of $ y = \log_{e}x $ is $ (0, \infty) $, and within this domain, the function is concave downwards, not upwards.

3. **Option 4: Concave downwards in $ (-\infty, \infty) $**
   - This is incorrect because the function $ y = \log_{e}x $ is only defined for $ x > 0 $. The interval $ (-\infty, \infty) $ includes negative values and zero, where the logarithmic function is not defined.

In summary, the function $ y = \log_{e}x $ is concave downwards in its domain $ (0, \infty) $, which supports Option 2 as the correct answer.