`" If " f(x) = |{:(3,,3x,,3x^(2)+2a^(2)),(3x,,3x^(2)+2a^(2),,3x^(3)+6a^(2)x),(3x^(2)+2a^(2),,3x^(3)++6a^(2)x,,3x^(4)+12a^(2)x^(2)+2a^(4)):}|` then

Options

A. f'(x)=0
B. y=f(x) is a straight line parallel to x-axis
C. `int_(0)^(2) f(x) dx=32 a^(4)`
D. none of these

Answer

A, B

Explanation

Applying `C_(3) to C_(3)-xC_(2) ,C_(2) to C_(2) -xC_(1)` we obtain
`Delta (x)= |{:(3,,0,,2a^(2)),(3x,,2a^(2),,4a^(2)x),(3x^(2)+2a^(2),,4a^(2)x,,6a^(2)x^(2)+2a^(4)):}|`
`=4a^(4) |{:(3,,0,,1),(3x,,1,,2x),(3x^(2)+2a^(2),,2x,,3x^(2)+a^(2)):}|`
Applying `C_(3) to C_(3) -xC_(2) `we get
`Delta (x) =4a^(4) |{:(3,,0,,1),(3x,,1,,x),(3x^(2)+2a^(2),,2x,,x^(2)+2a^(2)):}|`
Applying `C_(1) to C_(1)-3C_(3) ` we get
`Delta (x) =4a^(4) |{:(0,,0,,1),(0,,1,,x),(-4a^(2),,2x,,x^(2)+2a^(2)):}|=16a^(6)`