For the function f(x) = 4x^3 -36x^2 + 5: (a) find the critical numbers, (b) find the open intervals where the function is increasing or decreasing; and (c) apply the first derivative test to identify all relative extrema.

so we have those critical values, which gives us the intervals of (-∞, 0] , [0, 6] and [6, +∞).

where is it increasing or decreasing?  well, that's just a matter of checking a value on each region for the the first derivative, namely the first derivative test.

we can check for f(-1) = 12(-1)² - 72(-1),  is a positive value, increasing.

and check f(1) = 12(1)² - 72(1), is a negative value, decreasing.

and check f(7) = 12(7)² - 72(7), positive, so increasing.

check the picture below, and you can see there which are the minima or maxima.

cheater

step-by-step explanation:

step-by-step explanation:

let

we know that

the y-intercept is the y-coordinate of the point when the value of x is equal to zero

so

the y-coordinate of the point a is the y-intercept

the answer part a) is

the value of y-intercept is

part b) find the slope m

we know that

the value of the slope is equal to

substitute the values

therefore

the answer part b) is

the value of the slope is equal to

the general answer is

the slope is

the value of y-intercept is