Question

If $f\left(x\right)=1+2x^2+4x^4+6x^6+....+100x^{100}$ is a polynomial in a real variable $x$, then $f\left(x\right)$ has

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

Answer 1

The correct option is C only one minimum

$f\left(x\right)=1+2x^2+4x^4+6x^6+....+100x^{100}$

$f^{\prime }\left(x\right)=x(2^2+4^2.x^2+6^2.x^4+....+{100}^2.x^{98})$

For maxima or minima of $f\left(x\right)$

$f^{\prime }\left(x\right)=0\Rightarrow x(2^2+4^2.x^2+6^2.x^4+....+{100}^2.x^{98})=0$

$\Rightarrow x=0$ is only solution.

Also $f^{\prime \prime }\left(0\right)>0$

Hence $f\left(x\right)$ will attain only one minima.