Piece of cake!
Calculus
Level
4
If a,b,c,d,e are positive real numbers,such that a+b+c+d+e=8 and a² +b²+ c²+ d²+ e²=16,and range of e varies from 'm' to 'n'(m<n), then find value of m+5n.
The answer is 16.
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1 solution
As we know,
(¼)² (a+b+c+d)² ≤ ¼ (a² +b²+ c²+ d²) (Using Tchebycheff’s inequality) …(i)
where a+b+c+d+e=8
and a² +b²+ c²+ d²+ e²=16
Therefore, Eq.(i) reduces to (¼)² (8-e)²≤ ¼(16- e²)
⇒ 64+ e²-16e ≤ 4(16- e²)
⇒ 5 e²-16e ≤ 0
⇒ e(5e-16) ≤ 0
Using number line rule,
⇒ 0 ≤e ≤ 16/5
Thus, m=0 and n=16/5. So, m+5n=16.