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 . 

Is f(x) increasing, decreasing, or flat at ?

Recall that to find if a function is increasing or decreasing, we can use its first derivative. If f'(x) is positive, f(x) is increasing. If f'(x) is negative, f(x) is decreasing.

So, given:

We get

Then:

Therefore, f(x) is decreasing at the point, because f'(x) is negative.

Tell whether g(t) is increasing or decreasing on the interval [4,7]

To find increasing and decreasing, find where the first derivative is positive and negative. If g'(t) is positive, then g(t) is increasing and vice-versa.

Then,plug in the endpoints of [4,7] and see what you get for a sign.

So, since g'(t) is positive on the interval, g(t) is increasing.

To find the interval(s) on which the function is increasing, we must find the intervals on which the first derivative of the function is positive. 

The first derivative of the function is:

and was found using the rule

Now we must find the critical value, at which the first derivative is equal to zero:

Now, we make the intervals on which we look at the sign of the first derivative:

On the first interval the first derivative is positive, while on the second it is negative. Thus, the first interval is our answer, because over this range of x values, the first derivative is positive and the function is increasing. 

If  is continuous everywhere and always increasing (i.e.  for all ), then it must be true that after  has attained its root, it can never do so again because it can't "return" to the -axis. NOTE: this is not automatically true of functions that aren't continuous. As for the other choices, the possibility of at least having one more root is automatically false and a simple counterexample to the notion thay  has to have an inflection point is a simple increasing lines. It has a constant positive derivative, but possesses no upward or downward concavity and has no inflection points.

Take the first derivative of :

   by quotient rule

 is increasing whenever  is positive, that is, whenever both the numerator and denominator are of the same sign. The function  is certainly positive for all values of  greater than  because  and since 

is positive for all positive , it is increasing on the interval, too. It will never be negative. For the same reason, the numerator is always positive. With the numerator and denominator always positive everywhere on the given interval, the derivative is always positive and the function is always increasing. So for any interval of nonzero length within ,  is increasing.

NOTE: Interestingly the opposite of the choice  >  is also true.  on the entire interval because at , we have

. So the numerator is larger to begin with, and since:

  for all  (or any  for that matter), the derivative of the numerator is greater, too. This means the numerator will always be larger, so this condition coincides with the condition of  being positive.

To find the intervals on which the function is increasing, we must find the intervals where the first derivative is positive. To do this, we must find the first derivative, and find its critical values (at which the first derivative is equal to zero):

The derivative was found using the following rule:

Now, write the intervals of the function for which c is the upper and lower bound:

Note that at the critical value, the derivative is neither positive nor negative.

Now, we analyze the sign of the derivative within each interval; on the first interval, the derivative is always negative, but on the second interval, the first derivative is always positive. In other words, for this set of values -  -  the function is increasing. 

To determine the intervals on which the function is increasing, we must determine the intervals on which the first derivative of the function is positive. To start, we must find the first derivative:

The derivative was found using the following rule:

The first derivative is a positive constant, therefore the function is increasing on the entire domain, .