Question

Let $f\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+\dots .+a_n{x}^{2n}$ when $0<a_0<a_1<\dots \dots <a_n$ then $f\left(x\right)$ has

• A. No extremum
• B. Only one maximum
• C. Only one minimum
• D. Two maximums

Answer 1

The correct option is C Only one minimum

$f^{\prime }\left(x\right)=x(2a_1+4a_2{x}^2-----2n{a}_n{x}^{2n-2})$

$f^{\prime }\left(x\right)=0$ for $x=0$

$2a_1+4a_2{x}^2-----2n{a}_n{x}^{2n-2}>0$ for all $x$
$\therefore f^{\prime }\left(x\right)>0$ for all $x>0$
and $f^{\prime }\left(x\right)<0$ for all $x<0$
So $f^{\prime }\left(x\right)=0$ has only one root.
So $f\left(x\right)$ has only one minima.