Find the Range!

Algebra Level 5

If $a,b,c,d,e$ are positive real numbers, such that $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ , then $e \in [0,\dfrac p q]$

Note : If p=24, q= 49 , then the answer should be 2449

2 solutions

Nishant Rai
Jun 24, 2015

As we know $\large (\frac{a+b+c+d}{4})^2 \leq \frac{a^2+b^2+c^2+d^2}{4} .......(i)$

(using Tcheby cheff's Inequality )

Equation $(i)$ reduces to

$\large (\frac{8-e}{4})^2 \leq \frac{16-e^2}{4}........(ii)$

$\text{On solving (ii) } \rightarrow \large 0 \leq e \leq \frac{16}{5}$

what's the TCHEBY CHEFF'S INEQUALITY?

neelesh vij - 5 years, 6 months ago

Prakhar Bindal
Oct 1, 2016

Directly Apply Cauchy Schwarz Inequality on a,b,c,d to get desired result!