If $f(x)=x^{x}$, $x0$ is strictly increasing, then :

Question

If $f(x)=x^{x}$, $x>0$ is strictly increasing, then :

Accepted Answer

To determine when the function $ f(x) = x^x $ is strictly increasing for $ x > 0 $, we need to analyze the derivative of the function. The function $ f(x) = x^x $ can be rewritten using the exponential and logarithmic identities as $ f(x) = e^{x \ln x} $.

To find when $ f(x) $ is strictly increasing, we need to find the derivative $ f'(x) $ and determine where it is positive. Using the chain rule and the product rule, the derivative is:

$$ f'(x) = \frac{d}{dx}(e^{x \ln x}) = e^{x \ln x} \cdot \frac{d}{dx}(x \ln x). $$

Now, calculate $ \frac{d}{dx}(x \ln x) $:

$$ \frac{d}{dx}(x \ln x) = \ln x + x \cdot \frac{1}{x} = \ln x + 1. $$

Thus, the derivative becomes:

$$ f'(x) = e^{x \ln x} \cdot (\ln x + 1). $$

For $ f(x) $ to be strictly increasing, we require $ f'(x) > 0 $. Since $ e^{x \ln x} > 0 $ for all $ x > 0 $, the condition simplifies to:

$$ \ln x + 1 > 0. $$

Solving this inequality:

$$ \ln x > -1 $$

$$ x > e^{-1} $$

$$ x > \frac{1}{e}. $$

Therefore, the function $ f(x) = x^x $ is strictly increasing for $ x > \frac{1}{e} $, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2: $ x > e $**

This condition is too restrictive. While $ f(x) = x^x $ is indeed increasing for $ x > e $, it is also increasing for values of $ x $ between $ \frac{1}{e} $ and $ e $. Therefore, this option does not capture the entire range where the function is increasing.

- **Option 3: $ x > e+1 $**

This condition is even more restrictive than Option 2. The function $ f(x) = x^x $ is increasing well before $ x $ reaches $ e+1 $. Thus, this option excludes a significant portion of the domain where the function is increasing.

- **Option 4: $ x > 1 + \frac{1}{e} $**

Similar to Options 2 and 3, this condition is unnecessarily restrictive. The function starts increasing at $ x = \frac{1}{e} $, which is less than $ 1 + \frac{1}{e} $. Therefore, this option does not accurately describe the interval where the function is strictly increasing.

In conclusion, Option 1 is the correct answer because it accurately identifies the interval $ x > \frac{1}{e} $ as the range where the function $ f(x) = x^x $ is strictly increasing.