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Question Number 169397 by mokys last updated on 29/Apr/22

Answered by mr W last updated on 30/Apr/22

$$AC⩽AB+BC$$$$\sqrt{{(x+y)}^2+{(3+1)}^2}⩽\sqrt{x^2+3^2}+\sqrt{y^2+1^2}=5$$$${(x+y)}^2+{(3+1)}^2⩽25$$$${(x+y)}^2⩽9$$$$-3⩽x+y⩽3$$$${(x+y)}_{max}=3✓$$

Commented by mr W last updated on 30/Apr/22

Answered by cortano1 last updated on 30/Apr/22

$$f(x,y,\lambda )=x+y+\lambda (\sqrt{x^2+9}+\sqrt{y^2+1}-5)$$$$f_x^{\prime }=1+\lambda \left(\frac{x}{\sqrt{x^2+9}}\right)=0$$$$f_y^{\prime }=1+\lambda \left(\frac{y}{\sqrt{y^2+1}}\right)=0$$$$f_{\lambda }^{\prime }=\sqrt{x^2+9}+\sqrt{y^2+1}-5=0$$$$\Rightarrow \lambda =\lambda ;\frac{\sqrt{x^2+9}}{x}=\frac{\sqrt{y^2+1}}{y}$$$$\Rightarrow \frac{x^2+9}{x^2}=\frac{y^2+1}{y^2};1+\frac{9}{x^2}=1+\frac{1}{y^2}$$$$\Rightarrow x^2=9y^2\Rightarrow \{\begin{array}{l}x=3y\\ x=-3y\end{array};\sqrt{9y^2+9}+\sqrt{y^2+1}=5$$$$\Rightarrow 9y^2+9=26+y^2-10\sqrt{y^2+1}$$$$\Rightarrow 10\sqrt{y^2+1}=17-8y^2$$$$\Rightarrow 100y^2+100=64y^4-272y^2+289$$$$\Rightarrow \{\begin{array}{l}y=\pm \frac{3}{4}\\ y=\pm \frac{\sqrt{21}}{2}\left(notsolution\right)\end{array}$$$$\Rightarrow x+y=\{\begin{array}{l}3y+y=4y=\pm 3\\ -3y+y=-2y=\mp \frac{3}{2}\end{array}$$$$max=3$$

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