find the maximum and minimum value of f (x)=x^50-x^20 in tge interval (0, pi/2)

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$$\begin{gathered} \mathrm{We} \mathrm{have}, \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{50}-{\mathrm{x}}^{20} , \mathrm{x}\in \left(0,\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow \mathrm{f}\text{'}\left(\mathrm{x}\right)=50{\mathrm{x}}^{49}-20{\mathrm{x}}^{19} \\ =10{\mathrm{x}}^{19}\left(5{\mathrm{x}}^{30}-2\right) \\ \mathrm{For} \mathrm{maxima} \mathrm{or} \mathrm{minima} \mathrm{f}\text{'}\left(\mathrm{x}\right)=0 \\ \Rightarrow 10{\mathrm{x}}^{19}\left(5{\mathrm{x}}^{30}-2\right)=0 \\ \Rightarrow \mathrm{Either} {\mathrm{x}}^{19}=0 \\ \mathrm{or} \left(5{\mathrm{x}}^{30}-2\right)=0 \\ \mathrm{when} {\mathrm{x}}^{19}=0 \\ \mathrm{x}=0 \mathrm{but} \mathrm{x}\in \left(0,\frac{\mathrm{\pi }}{2}\right), \mathrm{hence} \mathrm{x}=0 \mathrm{will} \mathrm{not} \mathrm{be} \mathrm{considered}. \\ \mathrm{When} \\ \left(5{\mathrm{x}}^{30}-2\right)=0 \\ \Rightarrow {\mathrm{x}}^{30}=\frac{2}{5}=0.4 \\ \Rightarrow \mathrm{x}={\left(0.4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$30$}\right.} \\ \mathrm{As} \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=10{\mathrm{x}}^{19}\left(5{\mathrm{x}}^{30}-2\right) \\ \mathrm{Now} \mathrm{f}\text{'}\left(\mathrm{x}\right)>0 \mathrm{when} \mathrm{x}>{\left(0.4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$30$}\right.} \left(\mathrm{i}.\mathrm{e}. \mathrm{f}\text{'}\left(\mathrm{x}\right) \mathrm{increases}\right)\mathrm{and} \mathrm{f}\text{'}\left(\mathrm{x}\right)<0 \mathrm{when} \mathrm{x}<{\left(0.4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$30$}\right.}\left(\mathrm{i}.\mathrm{e}. \mathrm{f}\text{'}\left(\mathrm{x}\right) \mathrm{decreases}\right), \\ \mathrm{hence} \mathrm{f}\text{'}\left(\mathrm{x}\right) \mathrm{will} \mathrm{have} \mathrm{point} \mathrm{of} \mathrm{local} \mathrm{minima} \mathrm{at} \mathrm{x}={\left(0.4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$30$}\right.}. \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{50}-{\mathrm{x}}^{20} \\ ={\mathrm{x}}^{20}\left({\mathrm{x}}^{30}-1\right) \\ \Rightarrow \mathrm{f}\left(0.4^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$30$}\right.}\right)={\left(0.4\right)}^{\frac{20}{30}}\left\{{\left(0.4\right)}^{\frac{30}{30}}-1\right\} \\ ={\left(0.4\right)}^{\frac{2}{3}}\left\{{\left(0.4\right)}^{}-1\right\} \\ ={\left(0.4\right)}^{\frac{2}{3}}\left\{-0.6\right\} \\ \mathbf{\Rightarrow }\mathbf{f}{\left(\mathbf{x}\right)}_{\mathbf{min}}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{6}\left\{{\left(\mathbf{0}\mathbf{.}\mathbf{4}\right)}^{\frac{\mathbf{2}}{\mathbf{3}}}\right\}\mathbf{ }\left(\mathbf{Answer}\right) \\ \mathrm{Here} \mathrm{maximum} \mathrm{value} \mathrm{cannot} \mathrm{be} \mathrm{determined} \mathrm{as} \mathrm{we} \mathrm{do} \mathrm{not} \mathrm{know} \mathrm{number} \mathrm{just} \mathrm{before} \mathrm{x}=\frac{\mathrm{\pi }}{2}. \\ \mathrm{But} \mathrm{we} \mathrm{can} \mathrm{say} \mathrm{that} \mathrm{maximum} \mathrm{value} \mathrm{tends} \mathrm{towards} \mathrm{f}\left(\frac{\mathrm{\pi }}{2}\right) \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{20}\left({\mathrm{x}}^{30}-1\right) \\ \mathbf{\Rightarrow }\mathbf{f}{\left(\mathbf{x}\right)}_{\mathbf{max}}\mathbf{\to }{\left(\frac{\mathbf{\pi }}{\mathbf{2}}\right)}^{\mathbf{20}}\left\{{\left(\frac{\mathbf{\pi }}{\mathbf{2}}\right)}^{\mathbf{30}}\mathbf{-}\mathbf{1}\right\}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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