Graph the following function. Estimate the intervals on which the function is increasing or decreasing and any relative maxima or minima. \( f(x)=x^{2} \) Determine on which interval(s) \( f(x) \) is increasing. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) is increasing on the interval(s) \( \square \). (Type your answer in interval notation. Use a comma to separate answers as needed.) B. There is no interval on which the function \( f(x) \) is increasing.

To graph the function \( f(x) = x^2 \) and determine the intervals on which it is increasing or decreasing, we'll follow these steps: 1. **Graph the Function:** The graph of \( f(x) = x^2 \) is a parabola that opens upwards, with its vertex at the origin (0,0). The function is positive for all real numbers \( x \). 2. **Determine Increasing and Decreasing Intervals:** To find the intervals where the function is increasing or decreasing, we need to look at the first derivative of the function. The first derivative of \( f(x) = x^2 \) is \( f'(x) = 2x \). - When \( f'(x) > 0 \), the function is increasing. - When \( f'(x) < 0 \), the function is decreasing. Since \( f'(x) = 2x \), we have: - \( f'(x) > 0 \) when \( x > 0 \) - \( f'(x) < 0 \) when \( x < 0 \) Therefore, the function \( f(x) = x^2 \) is increasing on the interval \( (0, \infty) \) and decreasing on the interval \( (-\infty, 0) \). 3. **Relative Maxima and Minima:** The function \( f(x) = x^2 \) has a relative minimum at the vertex (0,0) because the function is increasing to the right of the vertex and decreasing to the left of the vertex. There is no relative maximum because the function continues to increase without bound as \( x \) approaches positive infinity. Now, let's fill in the answer boxes: A. The function \( f \) is increasing on the interval(s) \( (0, \infty) \). B. There is no interval on which the function \( f(x) \) is increasing. (This statement is incorrect based on our analysis, so we do not select this option.)