GATE Electrical 2015 Set 1 | Question: 2

Question

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?

• A. $f(a) . f(b)=0$
• B. $f(a) . f(b) < 0$
• C. $f(a) . f(b) > 0$
• D. $f(a) / f(b) \leq 0$

Answer 1

By the Intermediate Value Theorem (IVT):

• If $f(x)$ were continuous and $f(a)$ and $f(b)$ had opposite sign $(f(a).f(b)<0)$, then  IVT guarantees a root  exists in $(a,b)$.
• If  no roots exists in $[a,b]$, then $f(a)$ and $f(b)$ must the same sign both positive or both negative and cannot be zero (otherwise one end would be a root).
  
  So, $f(a).f(b) > 0 $ is correct statement.