We all know that the derivative function is used to judge the monotonicity of the original function. If the derivative function is greater than zero, the original function is increasing; if the derivative function is less than zero, the original function is decreasing. After finding the derivative function, if you continue to find the derivative function, that is, you can use it to judge the increase or decrease, as shown in the following figure:

Let's take a look at the application of quadratic derivation in solving the finale problem of college entrance examination mathematical function.

Richard 20 10 national volume 1, question 20, known function.

(i) If yes, the range of values to be found;

(ii) Proof that:

Let's look at the first question first. First of all, we can know the domain of the function, which is easy to get.

From the known, simplified.

This is the time to observe this inequality. Obviously, each term has a factor, and it is greater than zero, so both sides can be multiplied, so there is, there is.

, that is, when < 0, this is the increasing function in the interval; At that time,; while

So there is a maximum at, that is. Because it's the same sentence.

It should be said that the first question is not difficult, and most students can do it. Look at the second question.

To prove this, you only need to prove when it is.

Monotonicity known from above but used for analysis is blocked. We can try another derivative method, obviously, when

Next, we will analyze the situation when < >. Similarly, when < >, that is, increasing function is in the interval, then, at this time, it is increasing function, so Yi Yi is also established.

To sum up, it proves that.

Here is another solution for your reference and comparison.

Solution: (I), and then

The title is equivalent to.

Let's order then.

> when and >, >.

The first question is very routine, so let's go straight to the second question. First, we should construct a new function. If we can't think of this, there's nothing we can do. Then take the derivative, and the result is shown in the following table.

, continue to deduce.

The minimum value decreases and increases.

As can be seen from the above table, and

, known by >

>, so >, that is, the interval is an increasing function.

So there is >, and,

Therefore >, that is, when > and >, >.

Mathematics problems in senior high school usually give a derivative at the end, and most of them are quadratic. Many times, when you see a problem, you will know that it needs guidance. When you find that there are still unknown things in the result after the first derivative, and the extreme value is not clearly displayed, you can consider the second derivative. Of course, I asked for guidance three times. At this time, we should be very careful and observe the overall situation, otherwise it will be easy to make mistakes in the future.