Here we will study the function f (x) = e ^ x sin (x), where x ∈ [0, 2π]. a) determine where the function is decreasing and increasing. b) find all local maximam and minimam. does the absolute (global) maximam / minimam have? c) determine where f (x) curves up and down. also find any turning points.

we are given

(a)

Firstly, we will find critical numbers

so, we will find derivative

now, we can set it to 0

and then we can solve for x

we get

now, we can draw a number line and then locate these values

and then we can find sign of derivative on each intervals

increasing intervals:

Decreasing interval:

(b)

Local maxima:

It is the value of x where function changes from increasing to decreasing

so, local maxima is at

Local minima:

It is the value of x where function changes from decreasing to increasing

so, local minima is at

now, we will plug critical numbers and end values into original function

and we get

At x=0:

At :

At :

At :

Global maxima:

It is the largest value among them

so, we get

Global minima:

It is the largest value among them

so, we get

(c)

now, we can find second derivative

now, we can set it to 0

and then we can solve for x

so, we get

now, we  can draw number line and locate these values

and then we can find sign of second derivative on each intervals

concave up intervals:

Concave down intervals:

Turning points:

All values of x for which concavity changes

so, we get turning points at

t = 3

step-by-step explanation:

put t = 3 in this expression :

(3²-1)(3²-2)(3² - 3)(3² - 4) = 8×7×6×5 = 56×30 = 1680