The function f(x) =`x^2` is increasing in the interval

Options

A. (-1,1)
B. `(-infty,infty)`
C. `(0,infty)`
D. `(-infty,0)`

Answer

To determine the interval in which the function \( f(x) = x^2 \) is increasing, we follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) = x^2 \).

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

### Step 2: Set the derivative greater than zero
To find where the function is increasing, we need to set the derivative greater than zero:

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

### Step 3: Solve the inequality
Now, we solve the inequality \( 2x > 0 \):

\[ x > 0 \]

### Step 4: Determine the interval
The solution \( x > 0 \) indicates that the function \( f(x) = x^2 \) is increasing for all values of \( x \) greater than 0. Therefore, the interval in which the function is increasing is:

\[ (0, \infty) \]

### Conclusion
Thus, the function \( f(x) = x^2 \) is increasing in the interval \( (0, \infty) \).

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