Question

Let $f:[a,b]\to R$ be differentiable on $[a,b]$ and $k\in R.$ Let $f\left(a\right)=0=f\left(b\right).$ Also let $J\left(x\right)=f^{\prime }\left(x\right)+kf\left(x\right)$.Then

• A. $J\left(x\right)>0$ for all $x\in [a,b]$
• B. $J\left(x\right)<0$ for all $x\in [a,b]$
• C. $J\left(x\right)=0$ has atleast one root in $(a,b)$
• D. $J\left(x\right)=0$ through $(a,b)$

Answer 1

The correct option is C $J\left(x\right)=0$ has atleast one root in $(a,b)$

Let $g\left(x\right)=e^{kx}f\left(x\right)$
$f\left(a\right)=0=f\left(b\right)$
Hence function satisfies all the conditions of the Rolle's theorem so by Rolle's theorem there exists alteast one
$c\in (a,b)$ such that
$g^{\prime }\left(c\right)=0$
$g^{\prime }\left(x\right)=e^{kx}{f}^{\prime }\left(x\right)+k{e}^{kx}f\left(x\right)$
$g^{\prime }\left(c\right)=0$
$e^{kc}\left(f^{\prime }\right(c)+kf(c)=0$
$\Rightarrow f^{\prime }\left(c\right)+kf\left(c\right)=0$
for atleast one $c\in (a,b)$