GATE 2015 EE – Question 12

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 1 mark · Multiple choice

If a continuous function $f(x)$ does not have a root in the interval $[a, b]$, then which one of the following statements is TRUE?

$f(a) \cdot f(b) = 0$
$f(a) \cdot f(b) < 0$
$f(a) \cdot f(b) > 0$
$f(a)/f(b) \leq 0$

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Correct answer: (C) $f(a) \cdot f(b) > 0$

Explanation

A continuous function that changes sign must pass through zero. With no root in the interval, $f(a)$ and $f(b)$ have the same sign, so their product is positive.