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.

 is increasing when  is positive (above the -axis). This occurs on the intervals .

A function  is increasing if, for any ,  (i.e the slope is always greater than or equal to zero)

Function E is the only function that has this property. Note that function E is increasing, but not strictly increasing

To find the increasing intervals of a given function, one must determine the intervals where the function has a positive first derivative. To find these intervals, first find the critical values, or the points at which the first derivative of the function is equal to zero.

For the given function, .

This derivative was found by using the power rule 

. 

When set equal to zero, . Because we are only considering the open interval (0,5) for this function, we can ignore . Next, we look the intervals around the critical value , which are  and . On the first interval, the first derivative of the function is negative (plugging in values gives us a negative number), which means that the function is decreasing on this interval. However for the second interval, the first derivative is positive, which indicates that the function is increasing on this interval .