Question

Find the points of maximum and minimum and the intervals of monotonicity of the following function $f\left(x\right)=\frac{x^3}{(x^2+3)}$

Answer 1

$f\left(x\right)=\frac{x^3}{x^2+3}$

here $f^{\prime }\left(x\right)=\frac{(x^2+3)\frac{d}{dx}\left(x^3\right)-\frac{d}{dx}(x^2+3)\cdot x^3}{(x^2+3{)}^2}$

$=\frac{(x^2+3)\left(3x^2\right)-x^3\left(2x\right)}{(x^2+3{)}^2}$

$=\frac{3x^4+9x^2-2x^4}{(x^2+3)}=\frac{x^4+9x^2}{(x^2-3{)}^2}$

$=\frac{x^2(x^2+9)}{(x^2+3)}$ is always

Greater than zero for $x\in R$

So this function is always increasing because $f^{\prime }\left(x\right)>0$

for $x\in (-\infty ,\infty )$.