Which inequality?

Algebra Level 3

$\large \left(11 + \frac1{a^4} \right)^{\dfrac13} + \left(11 + \frac1{b^4} \right)^{\dfrac13}$

If $a$ and $b$ are positive numbers such that their sum is 1, find the minimum value of the expression above.

2 solutions

Kartik Sharma
Nov 15, 2015

It's really strange to see that there are now(included me) 100 solvers of this problem and not a single solution.

Here, I will use Jensen's inequality since the function $f(x) = \left(11+1/x^4\right)^{1/3}$ is convex.

Then by the inequality definition, $f(a) + f(b) \geq 2 f\left(\frac{a+b}{2}\right) = 2 f\left(\frac{1}{2}\right)$

$= 2 \left(11+2^4\right)^{1/3} = \boxed{6}$

Same solution here

Mohammed Imran - 1 year, 2 months ago

Yoogottam Khandelwal
Aug 25, 2016

Here, since we need to find the minimum value of the sum, we can assume that the least possible value is being added to 11.

for (1/a)^4 or (1/b)^4 to be minimum, a and b have to be at their maximum possible value.

since a + b = 1, we can say that both a & b will be maximum when a = b = (1/2)

putting a = b = (1/2) in the given equation, we get 6 as the answer