Find the maximum and minimum values of `f(x)=x^(50)-x^(20)` in the interval `[0,1]` .

Explanation

`f(x)=x^{50}-x^{20}` in interval `[0,1]`


`f^{prime}(x)=50 x^{49}-20 x^{19}` (for critical points) `f^{prime}(x)=0,` `50 x^{49}=20 x^{19}`


`x^{30}=frac{2}{5}` `x=pm(frac{2}{5})^{1 / 30}` Now, maximum value `f[(frac{2}{5})^{1 / 30}]=(frac{2}{5})^{5 / 3}-(frac{2}{5})^{2 / 3}`


`=(frac{2}{5})^{2 / 3}[(frac{2}{5})^{3}-1]`


`=-frac{117}{125}(frac{2}{5})^{2 /3}`


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


`=(frac{2}{5})^{2 / 3}[(frac{2}{5})^{3}-1]`


`=-frac{133}{125}(frac{2}{5})^{2}`