UPSC NDA II 2025 Mathematics Paper

Concept:The derivative of a constant function is zero. The absolute value function creates piecewise linear segments with different slopes.Explanation:Write $f(x)=|x-3|+|x-4|$ as a piecewise function. The critical points are $x=3$ and $x=4$. For $3\le x<4$, both absolute values combine to give $f(x)=1$, a horizontal line. At $x=3.5$, the function value is $1$ and it is constant in that interval. Therefore, the derivative, which measures the instantaneous rate of change, is $0$. There is no change in $y$ as $x$ changes within that interval.Answer:A. 0