120-x-1-x-3-x-5-max-

Question Number 206987 by hardmath last updated on 02/May/24

\frac{120}{∣ x - 1 ∣ + ∣ x - 3 ∣ + ∣ x - 5 ∣} \Rightarrow \max = ?

Answered by Frix last updated on 02/May/24

\frac{120}{f (x)} is \max \Rightarrow f (x) is \min

4 ⩽∣ x - 1 ∣ + ∣ x - 3 ∣ + ∣ x - 5 ∣

\Rightarrow Answer is \frac{120}{4} = 30

Commented by hardmath last updated on 02/May/24

thank you very much dear professor

Answered by A5T last updated on 03/May/24

∣ x - 3 ∣=∣ 3 - x ∣; ∣ x - 5 ∣=∣ 5 - x ∣; ∣ a ∣ + ∣ b ∣⩾∣ a + b ∣

∣ x - 1 ∣ + ∣ 3 - x ∣⩾∣ x - 1 + 3 - x ∣= 2 \dots (i)

S i m i l a r l y, ∣ x - 1 ∣ + ∣ 5 - x ∣⩾ 4 \dots (i i)

∣ x - 3 ∣ + ∣ 5 - x ∣⩾ 2 \dots (i i i)

(i) + (i i) + (i i i) \Rightarrow∣ x - 1 ∣ + ∣ x - 3 ∣ + ∣ x - 5 ∣⩾ 4

(E q u a l i t y w h e n x = 3)

\Rightarrow \frac{120}{∣ x - 1 ∣ + ∣ x - 3 ∣ + ∣ x - 5 ∣} ⩽ \frac{120}{4} = 30 .

Commented by hardmath last updated on 03/May/24

thank you so much dear professor