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. $f\left(a\right).f\left(b\right)=0$
• B. $f\left(a\right).f\left(b\right)<0$
• C. $f\left(a\right).f\left(b\right)>0$
• D. $f\left(a\right)/f\left(b\right)\le 0$

Answer 1

The correct option is C $f\left(a\right).f\left(b\right)>0$

If $f\left(x\right)$ is continuous and does not have a roots in $[a,b]$, then the curve $y=f\left(x\right)$ does not intersect co-ordinate axes.




Therefore the curve $y=f\left(x\right)$ completely lie either above $x-$axis ore below $x-$axis
Hence, $f\left(a\right)$ and $f\left(b\right)$ should have same sign.
$\therefore$$f\left(a\right)$.$f\left(b\right)>0$