Question

Let $P\left(x\right)=a_0+a_1{x}^2+a_2{s}^4+........+a_n{x}^{2n}$ be a polynomial in a real variable x with $0<a_0<a_1<a_2<....<a_{n^{\prime }}$ The function P(x) has

• A. neither a maximum nor a minimum
• B. only one maximum
• C. only one minimum
• D. only one maximum and only one minimum

Answer 1

Answer: (C)

only one minimum

$P\left(x\right)=a_0+a_1{x}^2+a_2{s}^4+........+a_n{x}^{2n}$
$P^{\prime }\left(x\right)=2a_{1}x+4a_2{x}^3+..............+2n{a}_n{x}^{2n-1}$
$\Rightarrow P^{\prime }\left(x\right)=x(2a_1+4a_2{x}^2+..............+2n{a}_n{x}^{2n-2})$
For extrema of $P\left(x\right)$
$P^{\prime }\left(x\right)=0\Rightarrow x(2a_1+4a_2{x}^2+..............+2n{a}_n{x}^{2n-2})=0$
$\Rightarrow x=0$ is only solution of this equation
Also $P^{\prime \prime }\left(0\right)>0$
Hence $x=0$ is only minima of $P\left(x\right)$