Question

Is `f(x)=e^x+cosx^2` increasing or decreasing at `x=pi/6`?

Answer 1

Increasing.

Explanation:

Find the sign of `f'(pi/6)`. If it's positive, the function is increasing. If it's negative, the function is decreasing at that point.

Find `f'(x)`.

`f'(x)=d/dx(e^x)+d/dx(cos(x^2))`

`d/dx(e^x)=e^x`

Use the chain rule to find the next derivative.

`d/dx(cos(u))=-u'sin(u)`

So,

`d/dx(cos(x^2))=-2xsin(x^2)`

Thus,

`f'(x)=e^x-2xsin(x^2)`

`f'(pi/6)~~1.405`

Since `f'(pi/6)>0`, the function is increasing when `x=pi/6`.

graph{e^x+cos(x^2) [-10.25, 15.05, -4.4, 8.26]}