Question

If $\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+a_n=0$. Then the function $f\left(x\right)=a_0{x}^n+a_1{x}^{n-1}+a_1{x}^{n-2}+.....+a_n$ has in (0,1)

• A. At least one zero
• B. At most one zero
• C. Only 3 zeros
• D. Only 2 zeros

Answer 1

The correct option is A At least one zero

$\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+...+\frac{a_n-1}{2}+an=0$
$f\left(x\right)=a_0{x}^n+a_1{x}^{n-1}+...+a_n=(1-x{)}^n$
$\int f\left(x\right)dx=\frac{a_0{x}^{n+1}}{n+1}+\frac{a_1{x}^n}{n}+...+a_{n}x$
$\therefore x=1$
$\frac{a_0}{n+1}+....+a_n=0$
$therefore$ At least one zero for $(1-x{)}^n$ in (0, 1)