If `f(x)=x^2+2b x+2c^2 and g(x)= -x^2-2c x+b^2` are such that min `f(x)> m a x g(x)` , then the relation between `b and c` is

Options

A. no real values b and c
B. `0 lt c lt sqrt(2) b`
C. `|c| lt sqrt(2)|b|`
D. `|c| gt sqrt(2)|b|`

Answer

D

Explanation

If `y = ax^(2) + bx + c`, then
`y_(min) = -((b^(2)-4ac)/(4a)), if a gt 0 and y_(max) = -((b^(2)-4ac)/(4a)), if a lt 0`
`therefore" "Min f(x) gt Max g(x)`
`rArr" "-((4b^(2)-8c^(2))/(4))gt -((4c^(2)+4b^(2))/(-4))`
`rArr" "(2c^(2) - b^(2)) gt (b^(2) + c^(2))`
`rArr" "c^(2) - 2b^(2) gt 0 rArr c^(2) - (sqrt(2)|b|)^(2) gt 0 rArr |c| gt sqrt(2)|b|`