If f(x) = ax + 3sinx + 4cosx is injective then prove a ∈( ,5] U5, ) - Maths - Relations and Functions

Question

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

Ishwarmani · Accepted Answer

Dear Student, Please find below the solution to the asked query: We have,fx=ax+3sinx+4cosx⇒f'x=a+3cosx-4sinx⇒f'x=a+535cosx-45sinxLet cosA=35, then sinA=45⇒f'x=a+5cosAcosx-sinAsinx⇒f'x=a+5cosA+xNow, -1≤cosA+x≤1⇒-5≤5cosA+x≤5⇒a-5≤a+5cosA+x≤5+a⇒a-5≤f'x≤5+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+a≥0 and a-5≥0⇒a≥-5 and a≥5So, a∈[5,∞)If f'x<0, thena-5≤0 and 5+a≤0⇒a≤5 and a≤-5So, a∈(-∞,-5]Hence, a∈(-∞,-5]∪[5,∞) Regards

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