COMEDK 2025 Afternoon Solved Paper

Concept:A function is monotonic on an interval if its derivative does not change sign. Here, the derivative changes from negative to positive, so the function is not monotonic overall.Explanation:We analyse $f(x)=x^2-x+1$ on $(0,1)$.Compute the derivative: $f'(x)=2x-1$.Set $f'(x)=0$ to find the critical point: $2x-1=0$ gives $x=\frac12$, which lies inside $(0,1)$.For $0 < x < \frac12$, choose $x=0.25$: $f'(0.25)=2(0.25)-1 = -0.5<0$, so $f$ is decreasing on $(0,\frac12)$.For $\frac12 < x<1$, choose $x=0.75$: $f'(0.75)=2(0.75)-1 = 0.5>0$, so $f$ is increasing on $(\frac12,1)$.Because the function decreases on one part and increases on another part of the interval, it is neither entirely increasing nor entirely decreasing on $(0,1)$.Thus, $f(x)$ is not monotonic on $(0,1)$.Answer:Option C – Neither increasing nor decreasing.