Question

Assertion :If $27a+9b+3c+d=0$, then the equation $f\left(x\right)=4a{x}^3+3b{x}^2+2cx+d=0$ has at least one real root lying between $(0,3)$. Reason: If $f\left(x\right)$ is continuous in $[a,b]$, derivable in $(a,b)$ such that $f\left(a\right)=f\left(b\right)$, then there exists at least one point $c\in (a,b)$ such that $f^{\prime }\left(c\right)=0$

• A. both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. both Assertion and Reason are incorrect

Answer 1

The correct option is A both Assertion and Reason are correct and Reason is the correct explanation for Assertion

If $f\left(x\right)$ is continuous in $[a,b]$, derivable in $(a,b)$ such that $f\left(a\right)=f\left(b\right)$, then there exists at least one point $c\in (a,b)$ such that $f^{\prime }\left(c\right)=0$
let $F^{\prime }\left(x\right)=f\left(x\right)$
i.e $F^{\prime }\left(x\right)=4a{x}^3+3b{x}^2+2cx+d$
integrating on both sides
$\Rightarrow F\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+e$
$\Rightarrow F\left(0\right)=e$
$\Rightarrow F\left(3\right)=81a+27b+9c+3d+e=3(27a+9b+3c+d)+e=e$ ($\because$ given that $27a+9b+3c+d=0$)
$\Rightarrow F\left(0\right)=F\left(3\right)$ then there exists at least one point in $(0,3)$ such that $f\left(x\right)=0$
$\therefore$ $f\left(x\right)$ has at least one real root lying between $(0,3)$.
Hence, both Assertion and Reason are correct and Reason is the correct explanation for Assertion.