Let `f(x)=x^3+a x^2+b x+5sin^2x` be an increasing function on the set `Rdot` Then find the condition on `a` and `b`.

Options

A. `a^2+3b+15 gt 0`
B. `a^2+3b+15 lt 0`
C. `a^2-3b-15 gt 0`
D. `a^2-3b-15 lt 0 `

Answer

A

Explanation

We have
`f(x) =-x^3+ax^2+bx+5/2 sin 2x`
`rArr f(x)=-3x^2+2ax+b+5 cos 2x`
For f(x) to be decreasing on R, we must have
`f(x) lt 0 " for all " x in R `
`rArr - 3x^2+2ax+b+5 cos 2x lt 0 " for all " x in R `
`rArr -3x^2 -2ax-b-5 gt 0 " for all " x in R `
`3x^2-2ax -b- 5 lt 0 "for all" x in R `
`rArr 4a^2 -4 xx 3 (-b-5) lt 0 " " [ because " Disc " lt 0 ]`
`rArr a^2 = 3b+5 lt 0 `