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Question Number 174267 by mnjuly1970 last updated on 28/Jul/22

$$$$$$\mathrm{Q}::\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}:$$$$$$$$\mathbb{M}{\mathrm{ax}}_{x\in [0,\frac{\pi }{2}]}\{\sin \left(x\right).{\cos }^3\left(x\right)\mid \}=?$$$$$$$$$$

Commented by kaivan.ahmadi last updated on 28/Jul/22

$$f^{\prime }\left(x\right)={cos}^{4}x-3{sin}^2{xcos}^{2}x=0$$$$\Rightarrow {cos}^{2}x({cos}^{2}x-3{sin}^{2}x)=0\Rightarrow$$$${cos}^{2}x=0\Rightarrow x=\frac{\pi }{2}$$$${cos}^{2}x-3{sin}^{2}x=0\Rightarrow {tan}^{2}x=\frac{1}{3}$$$$\Rightarrow tanx=\frac{\sqrt{3}}{3}\Rightarrow x=\frac{\pi }{6}$$$$f\left(0\right)=0$$$$f\left(\frac{\pi }{2}\right)=0$$$$f\left(\frac{\pi }{3}\right)=\frac{1}{2}\times {\left(\frac{\sqrt{3}}{2}\right)}^3=\frac{3\sqrt{3}}{16}$$$$Maxf=\frac{3\sqrt{3}}{16}$$

Commented by mnjuly1970 last updated on 28/Jul/22

$$thanksalotsir$$$$also...\frac{3\sqrt{3}}{16}typo$$

Commented by kaivan.ahmadi last updated on 28/Jul/22

$$anotherway$$$$a={sin}^{2}x,b={cos}^{2}x,a+b=1$$$$Max\left(f\right)=Max{\left({sin}^{2}x\right)}^{\frac{1}{2}}.{\left({cos}^{2}x\right)}^{\frac{3}{2}}=$$$$a^{\frac{1}{2}}{b}^{\frac{3}{2}}:\frac{a}{\frac{1}{2}}=\frac{b}{\frac{3}{2}}$$$$b=3a,a+b=1\Rightarrow a=\frac{1}{4},b=\frac{3}{4}$$$$Max\left(f\right)={\left(\frac{1}{4}\right)}^{\frac{1}{2}}{\left(\frac{3}{4}\right)}^{\frac{3}{2}}=$$$$\frac{1}{2}\sqrt{{\left(\frac{3}{4}\right)}^3}=\frac{1}{2}\times \frac{3\sqrt{3}}{8}=\frac{3\sqrt{3}}{16}$$

Commented by infinityaction last updated on 28/Jul/22

$$how$$$$\frac{a}{\frac{1}{2}}=\frac{b}{\frac{3}{2}}$$

Commented by mnjuly1970 last updated on 28/Jul/22

$$becausea+b=cte$$$$$$

Commented by kaivan.ahmadi last updated on 28/Jul/22

$$proposition:$$$$ifa,b>0anda+b=k>0\left(cte\right)$$$$\Rightarrow Max(a^m.b^n):\frac{a}{m}=\frac{b}{n}$$

Commented by Tawa11 last updated on 28/Jul/22

$$\mathrm{Great}\mathrm{sirs}$$

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