Question

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

• A. More than one minimum
• B. Exactly one minimum
• C. At least one maximum
• D. None of the above

Answer 1

The correct option is B

Exactly one minimum

Step 1: Given information

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

Step 2: Taking differentiation of given equation

$\mathrm{f}\text{'}\left(\mathrm{x}\right)=0+2\left(2\mathrm{x}\right)+2^2\left(4{\mathrm{x}}^3\right)+.....+2^{10}\left(20{\mathrm{x}}^{19}\right)$

$\mathrm{f}\text{'}\left(\mathrm{x}\right)=\mathrm{x}\left[2\left(2\right)+2^2\left(4{\mathrm{x}}^2\right)+....+2^{10}\left(20{\mathrm{x}}^{18}\right)\right]$

For maxima or minima, $f\text{'}\left(x\right)=0$.

The function $f\left(x\right)$ increases strictly for both $x>0$ and $x<0$ and has minimum value at $x=0$.

Hence, the correct option is option(B).