Minimum-2

Algebra Level 2

For positive real x x , find the minimum value of x 2 + 2 x x^2 +\dfrac{2}{x} .


The answer is 3.

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Jun Shin by Jun Shin , 18, Hong Kong

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2 solutions

Rishabh Jain
Feb 25, 2016

S = x 2 + 2 x \Large \mathfrak{S}=x^2+\dfrac{2}{x} = x 2 + 1 x + 1 x \Large =x^2+\dfrac{1}{x}+\dfrac{1}{x} ≥ 3 ( 1 ) 1 3 = 3 \Large \geq 3(1)^{\frac 13}=\boxed 3 (By A M ≥ G M AM\geq GM ). (For equality x=1) \color{#D61F06}{\textbf{(For equality x=1)}}

8 Helpful 0 Interesting 1 Brilliant 0 Confused
Jun Shin
Feb 2, 2016

y y = x 2 {x^2} + 2 / x 2/x =/(x^{2}/)+/(1/x/)+/(1/x/)
AM-GM inequality: /(x^{2}/)+/(1/x/)+/(1/x/)≥3√1 minimum is 3 when x=1

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Use \ instead of / for proper rendering.

Nihar Mahajan - 5 years, 3 months ago

Use x^{2} instead of x^2 for proper rendering.

Venkata Karthik Bandaru - 5 years, 4 months ago

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