Question

Is `f(x)=cotx-e^xtanx` increasing or decreasing at `x=pi/6`?

Answer 1

The function is decreasing.

Explanation:

To figure the increasing or decreasing nature of a function we can look at its derivative. If `f'(x)` is positive at the specified `x` value then the function is increasing otherwise if `f'(x)` is negative at the specified value then the function is decreasing.

If the `f'(x)=0` then we have a stationary point.

So, to differentiate the function:

`f'(x) = -csc^2(x) - e^xtanx-e^xsec^2(x)`

The product rule was used in the differentiation.

Now putting our value of `x` into our function:

`f'(pi/6)= -csc^2(pi/6)-e^(pi/6)tan(pi/6)-e^(pi/6)sec^2(pi/6)`

`=-4-e^(pi/6)(1/3+4/3)=-4-5/4e^(pi/6)<0`

Therefore as `f'(pi/6)` is negative our function must be decreasing as we can see from the graph of `f(x)` below.