An algebra problem by I Gede Arya Raditya Parameswara

Algebra Level 3

Let x x and y y be positive numbers such that x y = 9 xy=9 . What is the minimum value of 16 x + 4 y 16x+4y ?


The answer is 48.

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I Gede Arya Raditya Parameswara by I Gede Arya Raditya Para… , 17

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3 solutions

Fidel Simanjuntak
Apr 18, 2017

By AM-GM Inequality, we have

16 x + 4 y 2 ≥ ( 16 × 4 × x y = 16 × 4 × 9 ) \dfrac{16x+4y}{\color{#D61F06}2} \color{#333333}\geq \left( \color{#3D99F6}\sqrt{16 \times 4 \times xy} = \sqrt{16 \times 4 \times 9} \right)

16 x + 4 y ≥ 2 × 24 16x + 4y \geq \color{#D61F06} 2 \color{#333333} \times \color{#3D99F6} 24

16 x + 4 y ≥ 48 16x + 4y \geq 48

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Satyam Tripathi
Jan 17, 2017

Am gm inequality

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Using am gm

16x+4y=2×8×3 = 48

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I think that you'll need to specify that x , y x,y must be positive, for otherwise 16 x + 4 y = 16 x + 36 x 16x + 4y = 16x + \dfrac{36}{x} could be as negative as we want by letting x x go to 0 0 from the left.

Brian Charlesworth - 4 years, 5 months ago

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