The minimum value of \(5x^2 + \frac{5}{x^2}\) is? (where \(x \neq 0\))

The minimum value of the function 5 x 2 + x 2 5 ​ is 10 . This minimum occurs when x = 1 or x = − 1 . This was determined using the AM-GM inequality.
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Answered by Anonymous | 2025-07-04

To find the minimum value of the expression 5 x 2 + x 2 5 ​ , we will use the AM-GM inequality, which states that for any non-negative numbers a and b , the arithmetic mean is always greater than or equal to the geometric mean:
2 a + b ​ ≥ ab ​
Given the expression 5 x 2 + x 2 5 ​ , let us define:

a = x 2
b = x 2 1 ​

Notice that the expression can be rewritten as:
5 x 2 + x 2 5 ​ = 5 ( a + b )
Applying the AM-GM inequality on a and b , we get:
2 a + b ​ ≥ ab ​ = x 2 ⋅ x 2 1 ​ ​ = 1
Thus,
a + b ≥ 2
Multiplying both sides by 5 yields:
5 ( a + b ) ≥ 5 × 2 = 10
Therefore, the minimum value of 5 x 2 + x 2 5 ​ is 10 .
This minimum is achieved when a = b , meaning x 2 = x 2 1 ​ , which simplifies to x = ± 1 .
In conclusion, the minimum value of the expression 5 x 2 + x 2 5 ​ is 10 , attained when x = 1 or x = − 1 .

Answered by OliviaLunaGracy | 2025-07-06