Find the number of integral solution of x + 1/x6 + 2 > 1/4

Question

Find the number of integral solution of  x + 1/x² + 2 > 1/4

Answer 1

x + 1/x² + 2 - 1/4 > 0

⇒ x² - 4x - 2/x² + 2 < 0

⇒ [x -(2 + √6)][x -(2 -√6)] < 0

⇒ 2 - √6 < x < 2 + √6

See Fig.

 

Approximately, 0.4< x < 4.4

Therefore, integral values of x are 0, 1, 2, 3, 4. Hence, number of integral solutions is 5