# Is #f(x)=x-e^xsinx# increasing or decreasing at #x=pi/3#?

##### 1 Answer

Decreasing.

#### Explanation:

Use the sign, positive or negative, of the first derivative of a function to determine whether the function is increasing or decreasing:

- If
#f'(pi/3)<0# , then#f# is decreasing at#x=pi/3# . - If
#f'(pi/3)>0# , then#f# is increasing at#x=pi/3# .

So, we first must find **product rule**.

After applying the product rule and differentiating the initial

#f'(x)=1-sinxd/dx(e^x)-e^xd/dx(sinx)#

Recalling that

#f'(x)=1-e^xsinx-e^xcosx#

So, to determine whether the function is increasing or decreasing, we evaluate the derivative at

#f'(pi/3)=1-e^(pi//3)sin(pi/3)-e^(pi//3)cos(pi/3)approx-2.8927#

Since this is negative, the function is decreasing at

We can check a graph of

graph{x-e^xsinx [-9.78, 10.22, -6.75, 3.25]}

At