Question

If $P\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+.......+a_n{x}^{2n}$ be a polynomial in $x\in R$ with $0<a_1<a_2<.......<a_n$, then $P\left(x\right)$ has-

• A. no point of minimum
• B. only one point of minimum
• C. only two points of minimum
• D. None of these

Answer 1

The correct option is B only one point of minimum

Given $P\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+.......+a_n{x}^{2n}$
$P^{\prime }\left(x\right)=2x{a}_1+4a_2{x}^3+.......+2n{a}_n{x}^{2n-1}$
For max. or min. $P^{\prime }\left(x\right)=0$
$\Rightarrow$ $2x[a_1+2a_2{x}^2+.......+n.a_n{x}^{2n-2}]=0$
$\Rightarrow$ $x=0$
Now $P^{\prime \prime }\left(x\right)=2a_1+12a_2{x}^2+........+2n(2n-1)a_n{x}^{2n-2}$
$\Rightarrow$ $P^{\prime \prime }\left(x\right)=2a_1>0$ (given $a_1>0$)
Hence, $P\left(x\right)$ has only one minimum at $x=0$