If `alpha and beta` be the roots of the equation `x^(2) + px - 1//(2p^(2)) = 0`,
where `p in R`. Then the minimum value of `alpha^(4) + beta^(4)` is

Options

A. `2sqrt2`
B. `2 - sqrt2`
C. `2`
D. `2 + sqrt2`

Answer

4

Explanation

Here,
`alpha^(4) + beta^(4) = (alpha^(2) + beta^(2))^(2) - 2alpha^(2)beta^(2)`
`= {(alpha + beta)^(2) - 2alphabeta}^(2) - 2(alphabeta)^(2)`
`= (p^(2) + 1/p^(2))^(2) - (1)/(2p^(4))`
`= p^(4) + (1)/(2p^(4)) + 2`
`= (p^(2) - (1)/(sqrt2 p^(2)))^(2) + 2 + sqrt2 ge 2 + sqrt2`
Thus, the minimum value of `alpha^(4) + beta^(4)` is `2 + sqrt2`.