The required conditions for Intermediate Value Theorem include the function must be continuous and cannot equal . While there is a root at  for this particular continuous function, this cannot be shown using Intermediate Value Theorem. The function does not cross the  axis, thus eliminating that particular answer choice. The correct answer is “No, because .” Since one of the conditions for Intermediate Value Theorem is that cannot equal , by graphing  we can see that this requirement is not met.

his function is continuous (as it is a polynomial) and ; therefore, the required conditions for Intermediate Value Theorem are met. While there is a root at  for this function (as can be seen by graphing the polynomial), Intermediate Value Theorem does not state where this root will be exactly, nor does it state how many roots there might be. Thus, the conclusion that can be made by IVT is that there is a root for this polynomial located somewhere between  and .

First, determine the values of the function at the bounds. This will allow the correct implementation of the Intermediate Value Theorem.

Because the problem asks to analyze the interval  and , there must be a value , with . Because , by Intermediate Value Theorem there should be a number  between  and  that satisfies the required conditions. Therefore, “Yes, as shown by Intermediate Value Theorem” is the correct  answer.

If , only one root can be guaranteed (at ). 

If , then Intermediate Value Theorem can be applied twice, for  and .

This is true because for continuous functions, Intermediate Value Theorem states that a change in sign (ex: from positive to negative) of the function within an interval suggests a root (where the function crosses the  axis) at some point within that interval.

When you find the derivative of a function, you are finding the rate of change of the function.  By finding the derivative at a certain point you are actually finding the slope of the tangent line at that point, which tells you how much the function is changing at the given point.

To find the derivative of a function, you must find the derivative of each variable.  The derivative of a constant is .  Using our example of , to find the derivative we use the power rule.  The formula for the power rule is:

If  where is a constant and is an integer then 

So if  then .  This is the power rule in action.

To find the derivative we will apply the power rule to each variable.  Remember that the derivative of a constant is always .

Derivative of :

We multiply  by the coefficient () and then subtract one from the exponent giving us .

Derivative of :

 We multiply  by the coefficient () and then subtract one from the exponent giving us .

Derivative of :

 is just a constant, there is no variable here.  So the derivative of any constant is .

Putting this all together we have the derivative 

To find the rate of change at a given point first we must find the derivative of the function:

Now that we have found the derivative of the function, we need to plug in our given value, .

So the rate of change at  is .  This also means that the slope of the tangent line at this point is .

This is true.  When finding the derivative of a function you actually find a function that, when graphed, tells you when your original function is increasing or decreasing.  We can also find out if a function is increasing or decreasing at certain points by finding the derivative for specific points.  The slope of the tangent line that we find by doing this, tells us if our function is increasing (slope of the tangent line is positive) or decreasing (slope of the tangent line is negative).

If we were to graph this tangent line, we would see that the tangent line is increasing (it has a positive slope).  Since the tangent line at  is increasing, then our function must also be increasing at .