Question

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

• A. neither a maximum nor a minimum
• B. only one maximum
• C. only one minimum
• D. none of these

Answer 1

Answer: (C)

only one minimum

$\because$ $P\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+...+a_n{x}^{2n}$
$0<a_1<a_2<...<a_n$
$\Rightarrow$ $P^{\prime }\left(x\right)=2a_{1}x+4a_2{x}^3+6a_3{x}^5+...+2n{a}_n{x}^{2n-1}$
$=2x(a_1+2a_2{x}^2+3a_3{x}^3+...+n{a}_n{x}^{2n-2})$
$\therefore$ $P^{\prime }\left(x\right)=0$
$\Rightarrow$ $x=0$ & $P^{\prime \prime }\left(0\right)=2a_1>0$
$\therefore$ $x=0$ is onIy one minima.
Ans: C