The least value of α ∈ R for which, 4 α x 2 + 1 x   ≥ 1 , for all x > 0 , is

Question

The least value of &#945; &#8712; R for which, 4 &#945; x 2 + 1 x &#160; &#8805; 1 , for all x &#62; 0 , is

Aicharya · Accepted Answer

f x = 4 &#945; x 2 + 1 x ; x &#62; 0 &#8658; f ' x = 8 &#945; x - 1 x 2 = 8 &#945; x 3 - 1 x 2 &#8658; f x attains its minimum at x = 1 8 &#945; 1 3 f 1 8 &#945; 1 3 = 1 &#8658; 4 &#945; 1 8 &#945; 2 3 + 8 &#945; 1 3 = 1 &#8658; 3 &#945; 1 3 = 1 &#160; &#160; &#8658; &#160; &#160; &#945; = 1 27

The least value of &#945; &#8712; R for which, 4 &#945; x 2 + 1 x &#160; &#8805; 1 , for all x &#62; 0 , is

Solution

The least value of α ∈ R for which, 4 α x 2 + 1 x ≥ 1 , for all x > 0 , is