CUET UG 2025 — Mathematics Calculus

Given the function $f(x) = x^2 - x + 1$f(x)=x2−x+1

To find where the function is increasing or decreasing, find the derivative:

$$f'(x) = 2x - 1$$f′(x)=2x−1

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Setting $f'(x) = 0$f′(x)=0 to find the critical point:

$$2x - 1 = 0$$2x−1=0

$$2x = 1$$2x=1

$$x = \frac{1}{2}$$x=21​

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For $x < \frac{1}{2}$x<21​, test $x = 0$x=0:

$$f'(0) = 2(0) - 1 = -1$$f′(0)=2(0)−1=−1

Since $f'(0) < 0$f′(0)<0, the function is decreasing on $(-\infty, \frac{1}{2})$(−∞,21​)

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For $x > \frac{1}{2}$x>21​, test $x = 1$x=1:

$$f'(1) = 2(1) - 1 = 1$$f′(1)=2(1)−1=1

Since $f'(1) > 0$f′(1)>0, the function is increasing on $(\frac{1}{2}, \infty)$(21​,∞)

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The function is:

Decreasing on $(-\infty, \frac{1}{2})$(−∞,21​)

Increasing on $(\frac{1}{2}, \infty)$(21​,∞)