Question

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

Answer 1

`f(x)` is increasing at `pi/6`.

Explanation:

To figure out whether a function is increasing or decreasing at a certain point, we can take the function's derivative. If the derivative is positive, the function is increasing at that point, and if the derivative is negative, the function is decreasing at that point. If we take the derivative of this function using the quotient rule, we get `f'(x)=e^x(cos(x)sin^2(x)+sin^3(x)-sin(x)cos^2(x)+cos^3(x))`. Plugging in `pi/6`, we find that the derivative at that point `~~1.04`, so we know that the function is increasing.