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Question Number 208823 by Ismoiljon_008 last updated on 23/Jun/24

$$$$$$Ifa+b+c=15,thenfindthesmallestvalue$$$$oftheexpression\sqrt{a^2+1}+\sqrt{b^2+9}+\sqrt{c^2+16}.$$$$Helpplease$$

Answered by mr W last updated on 23/Jun/24

$$\sqrt{a^2+1^2}+\sqrt{b^2+3^2}+\sqrt{c^2+4^2}$$$$⩾\sqrt{{(a+b+c)}^2+{(1+3+4)}^2}$$$$=\sqrt{{15}^2+8^2}$$$$=17=minimum$$

Commented by Ismoiljon_008 last updated on 24/Jun/24

$$thanks$$$$$$$$$$

Commented by mr W last updated on 24/Jun/24

$$\mid \overrightarrow{A}\mid +\mid \overrightarrow{B}\mid +\mid \overrightarrow{C}\mid ⩾\mid \overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}\mid$$$$with\overrightarrow{A}=(a,1),\overrightarrow{B}=(b,3),\overrightarrow{C}=(c,4)$$

Commented by mr W last updated on 24/Jun/24

Answered by A5T last updated on 24/Jun/24

$${\left[E\right]}^2={[\sqrt{a^2+1}+\sqrt{b^2+9}+\sqrt{c^2+16}]}^2$$$$=a^2+b^2+c^2+1+9+16+2\sqrt{a^2+1}\sqrt{b^2+9}+$$$$2\sqrt{b^2+9}\sqrt{c^2+16}+2\sqrt{c^2+16}\sqrt{a^2+1}$$$$⩾a^2+b^2+c^2+26+2(ab+1\times 3)+2(bc+3\times 4)+$$$$2(ac+4\times 1)$$$$={(a+b+c)}^2-2ab-2bc-2ca+2ab+2ac+2bc+26$$$$+6+24+8={15}^2+8^2={17}^2$$$$E^2⩾{17}^2\Rightarrow E⩾17$$$$Equalitywhen(a,b,c)=(\lambda ,3\lambda ,4\lambda )\Rightarrow 8\lambda =15$$$$\Rightarrow \lambda =\frac{15}{8}\Rightarrow (a,b,c)=(\frac{15}{8},\frac{45}{8},\frac{15}{2})$$

Commented by A5T last updated on 24/Jun/24

$$\mid \mathit{u}\mid \mid \mathit{v}\mid ⩾\mathit{u}\cdot \mathit{v}$$$$\mathit{u}=(u_1,u_2),\mathit{v}=(v_1,v_2)$$$$\Rightarrow \sqrt{u_1^2+u_2^2}\sqrt{v_1^2+v_2^2}⩾u_1{v}_1+u_2{v}_2$$$$Equalityholdswhen(u_1,u_2)=(\lambda v_1,\lambda v_2)$$

Commented by Ismoiljon_008 last updated on 24/Jun/24

$$thank$$$$$$

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