The function $f(x) = x^2-x+1$ is

Increasing on $(\frac{1}{2},∞)$ and decreasing on $(-∞,\frac{1}{2})$

The correct answer is Option (2) → Increasing on $(\frac{1}{2},∞)$ and decreasing on $(-∞,\frac{1}{2})$

Given function: $f(x) = x^{2} - x + 1$

Compute derivative:

$f'(x) = 2x - 1$

Set $f'(x) = 0$ to find critical point:

$2x - 1 = 0 \Rightarrow x = \frac{1}{2}$

For $x < \frac{1}{2}$, $f'(x) < 0$ → function decreasing.

For $x > \frac{1}{2}$, $f'(x) > 0$ → function increasing.

Therefore:

Decreasing on $(-\infty, \frac{1}{2})$

Increasing on $(\frac{1}{2}, \infty)$

Correct option: Increasing on $(\frac{1}{2}, \infty)$ and decreasing on $(-\infty, \frac{1}{2})$.