If the parabola `y=ax^2+bx+c` has vertex at(4,2)and ` a in [1,3]` then the difference beteween the extreme value of abc is equal to

Options

A. 3600
B. 144
C. 3456
D. none of these

Answer

C

Explanation

The vertex of the parabola `y = ax^(2) + bx + c ` is at `(-(b)/(2a) , -(b^(2) -4ac)/(4a))`
`therefore -(b)/(2a) =4 ` and `-b^(2) -(4ac)/(4a) = 2`
`implies b = - 8a ` and `c - (b^(2))/(4a) = 2`
`implies b = -8a ` and `c =16a + 2 `
Now ,
f(a) = `abc = - 8a^(2) (16a + 2) = - 16(8a^(3) + a^(2))`
`(df(a))/(da) = -16 (24a^(2) + 2a)` ltLbrgt Clearly ,
`(d f(a))/(da) lt 0` for all a `in [1,3]`
`implies ` f(a) is decreasing on [1,3]
`therefore` Min value of f(a) = f(3) = `-16 (8 xx 3^(3) + 3^(2)) = - 3600`
Max. value of f (a) = f (1) = `-16(8 +1) = -144`
Hence , required difference = `-144 + 3600 = 3456`