Find the point at which the slope of the tangent of the function `f(x)=e^xcosx` attains minima, when `x in [0,2pi]dot`

Answer

`pi`

Explanation

Here `f(X) =e^(x)cosx`
`therefore f(X) =e^(x)cos x-e^(x) sin x= e^(x)(cosx-sinx)`
where f(X) is slope of tangent (which is to be minimized)
So let f(X) =g(x) Therefore
`g(x)=e^(x)(cosx-sinx)`
`therefore g(x) = e^(x) {cos x- sin x} +e^(x){sinx -cos xx}`
`=e^(x){-2 sinx }`
which is ve when x `in [pi,2pi]` and -ve when x `in [0,pi]`
Thus g(x) is decreasing in `(0,pi)` and increasing in `(pi,2pi)`
So at `x = pi` slope of tangent of the function f(x) attains minima.