Let \(f:[a, b] \rightarrow \mathbb{R}\) be differentiable on \([a, b] \& k \in \mathbb{R} .\) Let \(f(a)=0=f(b)\) Also let \(J(x)=f^{\prime}(x)+k f(x)\). Then

Question

\begin{aligned}
&\text { Let } f:[a, b] \rightarrow \mathbb{R} \text { be differentiable on }[a, b] \& k \in \mathbb{R} . \text { Let } f(a)=0=f(b) \\
&\text { Also let } J(x)=f^{\prime}(x)+k f(x) . \text { Then } \\
&\begin{array}{ll}
\text { (A) } J(x)>0 \text { for all } x \in[a, b] & \text { (B) } J(x)<0 \text { for all } x \in[a, b] \\
\text { (C) } J(x)=0 \text { has at least one root in }(a, b) & \text { (D) } J(x)=0 \text { through }(a, b)
\end{array} \\
&\text { }()
\end{aligned}

Answer 1

Ans : (C)
Hint : Let \(g(x)=k x f(x)\) which is continuous in [a, b] and differentiable in (a, b) \(g(a)=0=g(b))\)
\(\Rightarrow g^{\prime}(c)=0\) for same \(c \in(a, b)\) ( by Rolle's theorem)
\(\Rightarrow k f(c)+k c f^{\prime}(c)=0\)
\(\Rightarrow j(x)=0\) for same \(x=c \in(a, b)\)

Answer 2

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Answer 3

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