Let `f(x) =ax^(2) + bx + c and f(-1) lt 1, f(1) gt -1, f(3) lt -4 and a ne 0`, then

Options

A. `a gt 0`
B. `a lt 0`
C. sign of a cannot be determined
D. none of these

Answer

B

Explanation

We have, `f(-1) lt 1, f(1) gt -1 and f(3) lt -4`
`rArr" "a-b+c lt 1" "...(i)`
`rArr" "a+b+c gt -1" "...(ii)`
`rArr" "9a + 3b + c lt -4" "...(iii)`
From (ii), we have
`-a -b -c lt 1" "...(iv)`
Multiplying (i) by 3 and adding to (iii), we get
`12a + 4c lt -1 rArr 3a + c lt -(1)/(4)" "...(v)`
Multiplying (iv) by 3 and adding to (ii), we get
`6a - 2c lt -1 rArr 3a - c lt -(1)/(2)" "...(vi)`
Adding (v) and (vi), we get
`6a lt -(3)/(4) rArr a lt -(1)/(8) rArr a lt 0`.