Question

Find the maximum and minimum value $f\left(x\right)=x^{50}-x^{20}$ in the interval $[0,1]$.

Answer 1

$f\left(x\right)=x^{50}-x^{20}$ in interval $[0,1]$

$f^{\prime }\left(x\right)=50x^{49}-20x^{19}$ (for critical points)

$f^{\prime }\left(x\right)=0$

$50x^{49}=20x^{19}$

$x^{30}=\frac{2}{5}$

$x=\pm {\left(\frac{2}{5}\right)}^{1/30}$

Now, maximum value $f\left[{\left(\frac{2}{5}\right)}^{1/30}\right]={\left(\frac{2}{5}\right)}^{5/3}-{\left(\frac{2}{5}\right)}^{2/3}$

$={\left(\frac{2}{5}\right)}^{2/3}[{\left(\frac{2}{5}\right)}^3-1]$

$=-\frac{117}{125}{\left(\frac{2}{5}\right)}^{2/3}$

Minimum value, $f[-{\left(\frac{2}{5}\right)}^{1/30}]=-{\left(\frac{2}{5}\right)}^{5/3}+{\left(\frac{2}{5}\right)}^{2/3}$

$={\left(\frac{2}{5}\right)}^{2/3}[{\left(\frac{2}{5}\right)}^3-1]$

$=-\frac{133}{125}{\left(\frac{2}{5}\right)}^{2/3}$