How many critical points

First the derivative will not exist if there is division by zero in the denominator. So we need to solve,. However, these are NOT critical points since the function will also not exist at these points. Recall that in order for a point to be a critical point the function must actually exist at that point.

At this point we need to be careful. We can use the quadratic formula on the numerator to determine if the fraction as a whole is ever zero. So, we get two critical points. This will happen on occasion. Note as well that we only use real numbers for critical points. So, if upon solving the quadratic in the numerator, we had gotten complex number these would not have been considered critical points. In the previous example we had to use the quadratic formula to determine some potential critical points.

We know that sometimes we will get complex numbers out of the quadratic formula. Just remember that, as mentioned at the start of this section, when that happens we will ignore the complex numbers that arise. So far all the examples have not had any trig functions, exponential functions, etc.

The only critical points will come from points that make the derivative zero. We will need to solve,. There will be problems down the road in which we will miss solutions without this! Also make sure that it gets put on at this stage! Now divide by 3 to get all the critical points for this function. Notice that in the previous example we got an infinite number of critical points. Now, this looks unpleasant, however with a little factoring we can clean things up a little as follows,.

This function will exist everywhere, so no critical points will come from the derivative not existing. Determining where this is zero is easier than it looks.

We know that exponentials are never zero and so the only way the derivative will be zero is if,. Note that this function is not much different from the function used in Example 5. In this case the derivative is,. It is important to note that not all functions will have critical points! In this course most of the functions that we will be looking at do have critical points.

That is only because those problems make for more interesting examples. Do not let this fact lead you to always expect that a function will have critical points. The best answers are voted up and rise to the top.

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. How many critical points? Ask Question. Asked 9 years, 9 months ago. Active 9 years, 9 months ago. Viewed 3k times. Is there any way to prove this analytically? Arturo Magidin k 49 49 gold badges silver badges bronze badges. You can formally verify that the information from the graphing program is correct by appealing to the Intermediate Value Theorem and convexity.

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