GATE 2023 AE – Question 12
A monotonic continuous function $y=f(x)$ has only one root in $x_1<x<x_2$. Then
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Correct answer: (C) $f(x_1)f(x_2)<0$
Explanation
Let $f$ be continuous and monotonic on $[x_1,x_2]$ with exactly one root inside the interval.
- A continuous function that changes sign must cross zero (intermediate value theorem). A monotonic function crosses zero at most once, so having one root means $f$ goes from one sign to the opposite sign.
- Therefore $f(x_1)$ and $f(x_2)$ have **opposite signs** and neither is zero (the root lies strictly between $x_1$ and $x_2$).
$$f(x_1)\,f(x_2)<0 .$$
Option A would mean the same sign (no sign change, so no root). Option B would put the root at an end. Option D would say $f(x_1)=f(x_2)$, impossible for a strictly monotonic function. Answer **C**.