Question

Determine the points of maxima and minima of the function of $f\left(x\right)=\frac{1}{8}\log x-bx+x^2,x>0$ where $b>0$ is a constant.

Answer 1

$f\left(x\right)=\frac{1}{7}\log x-bx+x^2$

$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{8x}-b+2x=0$

$\Rightarrow 16x^2-8bx+1=0$

$\Rightarrow x=\frac{8b\pm \sqrt{64b^2-64}}{32}$

$=\frac{b}{4}\pm \frac{\sqrt{b^2-1}}{4}$

These are extremum points.