The right amount

Algebra Level 5

For variables p , q p,q and r r independent of x x , the equation x 4 + p x 3 + q x 2 + r x + 5 = 0 x^4 + px^3 + qx^2 + rx + 5 = 0 has four positive real roots of x x . Find the minimum value of p × r p \times r .


The answer is 80.

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Shashwat Avasthi by Shashwat Avasthi , 43, India

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3 solutions

Gian Sanjaya
Aug 22, 2015

By Vieta's formula, assuming a, b, c, and d are the real roots:

p r = ( a + b + c + d ) ( b c d + a c d + a b d + a b c ) pr = (a+b+c+d)(bcd+acd+abd+abc)

By Cauchy-Schwarz inequality:

p r ≥ ( 4 a b c d ) 2 = 16 a b c d pr \geq (4\sqrt{abcd})^2 = 16abcd

By Vieta's formula again:

p r ≥ 16 ∗ 5 = 80 pr \geq 16*5 = 80

When the equation is ( x − 5 4 ) 4 = 0 (x-\sqrt[4]{5})^4 = 0 , we get p r = 80 pr = 80 .

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Aakash Khandelwal
Aug 22, 2015

Take roots as a, b, c, d and apply AM>=HM

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I think this is the simplest way.....did it the same way

Samarth Agarwal - 5 years, 9 months ago
Sauditya Yo Yo
Aug 21, 2015

simple computation of vieta formulas and am-gm inequality

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Can you elaborate on it?

You meant Cauchy-Schwarz inequality? AM-GM is only for real positives, Cauchy-Schawrz is for all reals.

Gian Sanjaya - 5 years, 9 months ago

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Awww my bad the roots are positive reals...

Gian Sanjaya - 5 years, 9 months ago

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