If f(x) = ax + 3sinx + 4cosx is injective then prove a ∈(∞,5] U5, ∞)

Dear Student,
Please find below the solution to the asked query:

We have,fx=ax+3sinx+4cosxf'x=a+3cosx-4sinxf'x=a+535cosx-45sinxLet cosA=35, then sinA=45f'x=a+5cosAcosx-sinAsinxf'x=a+5cosA+xNow, -1cosA+x1-55cosA+x5a-5a+5cosA+x5+aa-5f'x5+a       .....iAs, fx is a injective funtionSo, f'x is either strictly increasing or strictly decreasing functioni.e. f'x>0 or f'x<0         .....iiFrom i and ii, we getIf f'x>0, then5+a0 and a-50a-5 and a5So, a[5,)If f'x<0, thena-50 and 5+a0a5 and a-5So, a(-,-5]Hence, a(-,-5][5,)
 
 
Regards

  • 12
What are you looking for?