Question

If `f(x)=x^2+2bx+2c^2`, `g(x)=-x^2-2cx+b^2` and `minf(x)>maxg(x)` then
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 1

Correct Answer - D
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|`