Question

Let $f\left(x\right)=1+2x^2+2^2{x}^4+.....+2^{10}{x}^{20}$. Then f(x) has

• A. more than one minimum
• B. exactly one minimum
• C. at least one maximum
• D. none of these

Answer 1

The correct option is B exactly one minimum

$f^{\prime }\left(x\right)=2.2x+2^2.4x^3+.....+2^{10}.20x^{19}$

$\Rightarrow f^{\prime }\left(x\right)=x[2.2+2^2.4x^2+.....+2^{10}.20x^{18}]$

For maxima or minima, $f^{\prime }\left(x\right)=0$

$\Rightarrow x=0$

The function increases strictly for both $x>0$ and $x<0$ and has minimum value at $x=0$

$\therefore$ Correct option is B