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}`

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]`

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