Prove that if `2a_0 ^2<15 a ,` all roots of `x^5-a_0x^4+3a x^3+b x^2+c x+d=0` cannot be real. It is given that `a_0,a ,b ,c ,d in Rdot`

Explanation

Let
`f(x) = x^(5) - a_(0) x^(4) + 3ax^(3) + bx^(2) + cx + d`
`therefore f'(x) = 5x^(4) - 4a_(0)x^(3) + 9ax^(2) + 2bx + c`
`rArr f''(x) = 20x^(3) - 12a_(0)x^(2) + 18ax^(2) + 2b `
`rArr ` f'''(x) = 60x^(2) - 24a_(0)x + 18a`
`rArr f''''(x) = 6(10x^(2) - 4a_(0)x + 3a)` ...(1)
from Roll's theorem, we know that between any two toots of the equation `f(x) = 0`, there exists at least one root of `f'(x) = 0` ,
Discriminant of polynomial (1) is
`D - 16_(0)^(2) - 4xx10xx3a`
`rArr D - 8(2a_(0)^(2) - 15a)lt [As 2a_(0)^(2) - 15a lt 0`given ]
Hence, `f'''(x) = 0` has non-real roots.
Therefore, `f''(x) = 0` has not all real roots.
With similar reasoning all the roots of `f'(x) = 0 and f(x) = 0` are not all real.