inequalities works

Algebra Level 2

what is the minimal value of the expression:

9 x 2 + y 2 + 1 x 2 + 1 16 y 2 ? 9x^2+y^2+\dfrac{1}{x^2}+\dfrac{1}{16y^2}?


The answer is 6.5.

Antonio Tagliente by Antonio Tagliente , 23, Italy

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Applying the Cauchy-Schwarz inequality to the quadruples:

A = ( 3 x , y , 1 x , 1 4 y ) A=(3x,y,\frac{1}{x},\frac{1}{4y}) B = ( 1 x , 1 4 y , 3 x , y ) B=(\frac{1}{x},\frac{1}{4y},3x,y)

we get: ∑ a × ∑ b ≥ ( 6 + 1 2 ) 2 \sum a \times \sum b \geq(6+\frac{1}{2})^2

so the expression is at least 6.5 6.5

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Complete the squares keeping in mind that each term is nonnegative

2 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...