Find the absolute extrema if they exist, as well as all values of x where they occur, for the function f(x)= x+e^-3x on the domain [-2,2]

The absolute extrema of the function on the domain are:

Absolute minimum:

Absolute maximum:

Step-by-step explanation:

Let with interval in , we can check if the absolute extrema exist by applying First and Second Derivative Test, otherwise we must evaluate the function at lower and upper bounds.

First, we obtain the first derivative of the function, which is later equalized to zero and solved for :

Second, we evaluated the second derivative of given function at result above:

(Absolute minimum)

The value of the absolute minimum is:

x = 0.366

Third, we evaluate the function at each bound in the search for absolute maximum:

x = -2

x = 2

The absolute extrema of the function on the domain are:

Absolute minimum:

Absolute maximum: