1) let f(x)=6x+6/x. find the open intervals on which f is increasing (decreasing). then determine the x-coordinates of all relative maxima (minima).

let f(x)=6−4/x+2/x^2. find the open intervals on which f is increasing (decreasing). then determine the x-coordinates of all relative maxima (minima).

let f(x)=(8x^2)/(x−4). find the open intervals on which f is increasing (decreasing). then determine the x-coordinates of all relative maxima (minima).

let f(x)=6(x−2)^(2/3) +4. find the open intervals on which f is increasing (decreasing). then determine the x-coordinates of all relative maxima (minima).

let f(x)=8√x −6x for x> 0. find the open intervals on which f is increasing (decreasing). then determine the x-coordinates of all relative maxima (minima).

1) increasing on (-∞,-1] ∪ [1,∞), decreasing on [-1,0) ∪ (0,1]

is local maximum, is local minimum

2) increasing on [1,∞), decreasing on (-∞,0) ∪ (0,1]

is absolute minimum

3) increasing on (-∞,0] ∪ [8,∞), decreasing on [0,4) ∪ (4,8]

is local maximum, is local minimum

4) increasing on [2,∞), decreasing on (-∞,2]

is absolute minimum

5) increasing on the interval (0,4/9], decreasing on the interval [4/9,∞)

is local minimum, is absolute maximum

Step-by-step explanation:

To find minima and maxima the of the function, we must take the derivative and equalize it to zero to find the roots.

1)

and

So, the roots are and

The function is increasing on the interval (-∞,-1] ∪ [1,∞)

The function is decreasing on the interval [-1,0) ∪ (0,1]

is local maximum, is local minimum.

2)

and

So the root is

The function is increasing on the interval [1,∞)

The function is decreasing on the interval (-∞,0) ∪ (0,1]

is absolute minimum.

3)

and

So the roots are and

The function is increasing on the interval (-∞,0] ∪ [8,∞)

The function is decreasing on the interval [0,4) ∪ (4,8]

is local maximum, is local minimum.

4)

has no solution and is crtitical point.

The function is increasing on the interval [2,∞)

The function is decreasing on the interval (-∞,2]

is absolute minimum.

5) for

So the root is

The function is increasing on the interval (0,4/9]

The function is decreasing on the interval [4/9,∞)

is local minimum, is absolute maximum.

the entire expression should be subtracted from 30. to subtract the quantity, distribute the negative sign, which changes the signs on the terms. so, the –2 changes to positive and gets added to 30 instead of subtracted from it.

hope this ! ^w^