Question

If $\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+......+\frac{a_{n-1}}{2}+a_n=0$ then $a_0{x}^n+a_1{x}^{n-1}+.......+......+a_{n-1}x+a_n=0$ has

• A. no solution in $(0,1)$
• B. at least one solution in $(0,1)$
• C. exactly one solution in $(0,1)$
• D. at least one solution in $(2,3)$

Answer 1

The correct option is B at least one solution in $(0,1)$

$f\left(x\right)=a_0{x}^n+a_1{x}^{n-1}\cdots +a_{n-1}x+an$
Let,

$g\left(x\right)={\int }_0^{x}f\left(x\right)=\frac{a_0{x}^{n+1}}{n+1}+\frac{a_1{x}^n}{n}$ $\dots \frac{a_{n-1}{x}^2}{2}+a_{n}x$
$g^{\prime }\left(x\right)=f\left(x\right)$
$g\left(x\right)$ is zero for $x=1$ and $x=0$
Therefore,

$g^{\prime }\left(x\right)=f\left(x\right)$ must be zero for at least one value between $0$ and $1$