If-a-b-c-15-then-find-the-smallest-value-of-the-expression-a-2-1-b-2-9-c-2-16-Help-please-

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

I f a + b + c = 15, t h e n f i n d t h e s m a l l e s t v a l u e

o f t h e e x p r e s s i o n \sqrt{a^{2} + 1} + \sqrt{b^{2} + 9} + \sqrt{c^{2} + 16} .

H e l p p l e a s e

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 = m i n i m u m

Commented by Ismoiljon_008 last updated on 24/Jun/24

t h a n k s

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

∣ \vec{A} ∣ + ∣ \vec{B} ∣ + ∣ \vec{C} ∣⩾∣ \vec{A} + \vec{B} + \vec{C} ∣

w i t h \vec{A} = (a, 1), \vec{B} = (b, 3), \vec{C} = (c, 4)

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

Answered by A5T last updated on 24/Jun/24

{[E]}^{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 (a b + 1 \times 3) + 2 (b c + 3 \times 4) +

2 (a c + 4 \times 1)

= {(a + b + c)}^{2} - 2 a b - 2 b c - 2 c a + 2 a b + 2 a c + 2 b c + 26

+ 6 + 24 + 8 = 15^{2} + 8^{2} = 17^{2}

E^{2} ⩾ 17^{2} \Rightarrow E ⩾ 17

E q u a l i t y w h e n (a, b, c) = (λ, 3 λ, 4 λ) \Rightarrow 8 λ = 15

\Rightarrow λ = \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

∣ u ∣∣ v ∣⩾ u \cdot v

u = (u_{1}, u_{2}), 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}

E q u a l i t y h o l d s w h e n (u_{1}, u_{2}) = (λ v_{1}, λ v_{2})

Commented by Ismoiljon_008 last updated on 24/Jun/24

t h a n k