Set of values of b for which local extrema of the function f(x) are positive where `f(x)=(2)/(3)a^(2)x^(3)-(5a)/(2)x^(2)+3x+b` and maximum occurs at `x=(1)/(3)` is -

Options

A. `(-4,oo)`
B. `(-3//8,oo)`
C. `(-10,3//8)`
D. None of these

Answer

2

Explanation

`f'(x)=2a^(2)x^(2)-5ax+3=(ax-1)(2ax-3)=0`
`x=(1)/(a),(3)/(2a)`
`"If a" gt 0 "then local maxima occurs at" x=(1)/(a)` and
`"minima at" x=(3)/(2a)`
`because "maxima occured" x=(1)/(a)=(1)/(2)rArra=3`, minima
`"occured" x=(3)/(2a)=(1)/(2)`
`f((1)/(2))gt0rArr(2)/(3)xx9xx(1)/(8)-(5)/(2)xx3xx(1)/(4)+3(1)/(2)+bgt0,`
`(6-15+12)/(8)+bgt0`
`rArr(3)/(8)+bgt0rArrbgt-(3)/(8)`
If a `lt` 0 then maxima shall occur at `x=(3)/(2)a` and
manima `x=(1)/(2a)`
`(3)/(2a)=(1)/(3)rArra=(9)/(2)gt "0 not admissible".`
Hence `bgt-(3)/(8)`.