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. One maximum and one minimum

Answer 1

The correct option is C Only one minimum

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

$f^{{}^{\prime }}\left(x\right)=0+4x+4^2.x^3+6^2.x^5+.....{100}^2{x}^{99}$

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

which is always positive never be zero.

$f^{{}^{\prime }}\left(x\right)=0[\because f^{{}^{\prime }}(x)=0,\text{only if}x=0]$

At x = 0 , we get minima.

So f(x) has only one minimum.