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?
1. \(f\left( a \right).f\left( b \right) = 0\)
2. \(f\left( a \right).f\left( b \right) < 0\)
3. \(f\left( a \right).f\left( b \right) > 0\)
4. \(f\left( a \right)/f\left( b \right) \le 0\)

Answer 1

Correct Answer - Option 3 : \(f\left( a \right).f\left( b \right) > 0\)

We know that, (Intermediate value theorem)

If \(f\left( a \right)\;f\left( b \right) < 0\) then f(x) has at least one root in (a, b)

Since there is no root in [a, b] this implies f(a) and f(b) are of same sign 

i.e. either they both are positive or they both are negative

In both cases \(f\left( a \right)f\left( b \right) > 0\)