Question

Is `f(x)=sinx/x` increasing or decreasing at `x=pi/3`?

Answer 1

Decreasing.

Explanation:

To determine if a function is increasing or decreasing at a point, use the function's derivative:

• If `f'(a)<0`, then `f` is decreasing at `x=a`.
• If `f'(a)>0`, then `f` is increasing at `x=a`.

So, we first must find the derivative of `f`. To do so, we will have to use the quotient rule. Application of the quotient rule shows that

`f'(x)=(xd/dx(sinx)-sinxd/dx(x))/x^2`

`=(xcosx-sinx)/x^2`

So, to determine if `f` is increasing or decreasing at `x=pi/3`, find `f'(pi/3)` and see if it is positive or negative.

`f'(pi/3)=(pi/3cos(pi/3)-sin(pi/3))/(pi/3)^2`

`=(pi/3(1/2)-sqrt3/2)/(pi^2/9)approx-0.3123`

Since this is `<0`, the function is decreasing at `x=pi/3`.

We can check a graph of `f` `(`note that `pi/3approx1.0472)`.

graph{sinx/x [-3.945, 4.825, -1.568, 2.817]}