Is the function `f(x)=x^(2), x in R` increasing?

Answer

To determine if the function \( f(x) = x^2 \) is increasing for all \( x \in \mathbb{R} \), we will follow these steps:

### Step 1: Find the derivative of the function
To analyze whether the function is increasing, we first need to find its derivative. The derivative of \( f(x) = x^2 \) is calculated as follows:

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

### Step 2: Analyze the sign of the derivative
Next, we need to determine when the derivative \( f'(x) = 2x \) is greater than or equal to zero. This will help us identify the intervals where the function is increasing.

\[ f'(x) \geq 0 \implies 2x \geq 0 \]

Dividing both sides by 2 (which does not change the inequality since 2 is positive):

\[ x \geq 0 \]

### Step 3: Determine the intervals of increase
From the inequality \( x \geq 0 \), we can conclude that the function \( f(x) = x^2 \) is increasing for all \( x \) in the interval \( [0, \infty) \).

### Step 4: Check the behavior for negative values
Now, we check the behavior of the function for \( x < 0 \):

\[ f'(x) < 0 \text{ for } x < 0 \]

This indicates that the function is decreasing for all negative values of \( x \).

### Conclusion
Thus, the function \( f(x) = x^2 \) is not increasing for all \( x \in \mathbb{R} \). It is increasing only on the interval \( [0, \infty) \) and decreasing on \( (-\infty, 0) \).

### Final Answer
The function \( f(x) = x^2 \) is not increasing for all real numbers \( x \).

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