To find the intervals on which the function is decreasing, we must find the intervals on which the first derivative of the function is negative.

To start, we must find the first derivative:

The derivative was found using the following rule:

Next, we must find the critical values, the values at which the first derivative is equal to zero:

Note that the first derivative was simplified using a factoring by grouping.

With the critical values, we can now make our intervals:

Note that at the endpoints of the intervals the first derivative is neither positive nor negative.

To check the sign of the first derivative on the intervals, simply plug in any point on each interval into the first derivative function and see the sign. On the first interval, the first derivative is negative, on the second, it is positive, on the third, it is negative, and on the fourth, it is positive. Thus, the intervals on which the function is decreasing are .

To find the intervals on which the function is decreasing, we must find the intervals on which the first derivative of the function is negative.

The first derivative of the function is

and was found using the following rule:

Next, we must find the critical values, or values at which the first derivative of the function is equal to zero:

Now, using the critical values, we can create the intervals:

Note that at the endpoints of the intervals, the first derivative is neither positive nor negative.

To determine the sign of the first derivative on each interval, simply plug in any value on each interval into the first derivative function and check the sign. On the first interval, the first derivative is positive, on the second interval, it is negative, and on the third, it is positive. Thus, the interval on which the function is decreasing is .

Is the function g(x) increasing or decreasing on the interval ?

To tell if a function is increasing or decreasing, we need to see if its first derivative is positive or negative. let's find g'(x)

Recall that the derivative of a polynomial can be found by taking each term, multiplying by its exponent and then decreasing the exponent by 1.

Next, we need to see if the function is positive or negative over the given interval.

Begin by finding g'(-5)

So, g'(-5) is negative, but what about g'(0)?

Also negative, so our answer is:

Decreasing, because g'(x) is negative on the interval .

We will use the tangent line slope to ascertain the increasing / decreasing of f(x). To this end, let us begin by taking the first derivative of f(x):

f'(x) = x² + 5x – 14

Solve for the potential relative maxima and minima by setting f'(x) to 0 and solving:

x² + 5x – 14 = 0; (x – 2)(x + 7) = 0

Potential relative maxima / minima: x = 2, x = –7

We must test the following intervals: (–∞, –7), (–7, 2), (2, ∞)

f'(–10) = 100 – 50 – 14 = 36

f'(0) = –14

f'(10) = 100 + 50 – 14 = 136

Therefore, the equation increases on (–∞, –7) and (2, ∞)

To find when a function is increasing, you must first take the derivative, then set it equal to 0, and then find between which zero values the function is positive.

First, take the derivative:

Set equal to 0 and solve:

Now test values on all sides of these to find when the function is positive, and therefore increasing. I will test the values of -6, 0, and 2.

Since the values that are positive is when x=-6 and 2, the interval is increasing on the intervals that include these values. Therefore, our answer is:

To find increasing and decreasing intervals, we need to find where our first derivative is greater than or less than zero. If our first derivative is positive, our original function is increasing and if g'(x) is negative, g(x) is decreasing. 

Begin with:

If we plug in any number from 3 to 6, we get a positve number for g'(x), So, this function must be increasing on the interval {3,6}, because g'(x) is positive. 

To find out if a function is increasing or decreasing, we need to find if the first derivative is positive or negative on the given interval.

So starting with:

We get:

 using the Power Rule .

Find the function on each end of the interval.

So the first derivative is positive on the whole interval, thus g(t) is increasing on the interval. 

A function is increasing on an interval if for every point on that interval the first derivative is positive.

So we need to find the first derivative and then plug in the endpoints of our interval.

Find the first derivative by using the Power Rule 

Plug in the endpoints and evaluate the function.

Both are positive, so our function is increasing on the given interval.

The first step is to find the first derivative.

Remember that the derivative of 

Next, find the critical points, which are the points where  or undefined. To find the  points, set the numerator to , to find the undefined points, set the denomintor to . The critical points are  and 

The final step is to try points in all the regions  to see which range gives a positive value for .

If we plugin in a number from the first range, i.e , we get a negative number.

From the second range, , we get a positive number.

From the third range, , we get a negative number.

From the last range, , we get a positive number.

So the second and the last ranges are the ones where  is increasing.