When do critical points occur

We will discuss the occurrence of local maxima and local minima of a function. In fact, these points are crucial to many questions related to optimization problems. We will discuss these problems in later pages. A function f x is said to have a local maximum at c iff there exists an interval I around c such that.

Using the definition of the derivative, we can easily show that: If f x has a local extremum at c , then either. Therefore the conditions. If a point is not in the domain of the function then it is not a critical point. Note as well that, at this point, we only work with real numbers and so any complex numbers that might arise in finding critical points and they will arise on occasion will be ignored.

There are portions of calculus that work a little differently when working with complex numbers and so in a first calculus class such as this we ignore complex numbers and only work with real numbers.

Calculus with complex numbers is beyond the scope of this course and is usually taught in higher level mathematics courses. The main point of this section is to work some examples finding critical points. Now, our derivative is a polynomial and so will exist everywhere. So, we must solve. They are,. This will allow us to avoid using the product rule when taking the derivative.

We will need to be careful with this problem. When faced with a negative exponent it is often best to eliminate the minus sign in the exponent as we did above. So, getting a common denominator and combining gives us,.

This negative out in front will not affect the derivative whether or not the derivative is zero or not exist but will make our work a little easier. Now, we have two issues to deal with. 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. Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema. The extreme value theorem: If f x is continuous in a closed interval I, then f x has at least one absolute maximum and one absolute minimum in I.

Occurrence of absolute maxima: If f x is continuous in a closed interval I, then the absolute maximum of f x in I is the maximum value of f x on all local maxima and endpoints on I. Occurrence of absolute minima: If f x is continuous in a closed interval I, then the absolute minimum of f x in I is the minimum value of f x on all local minima and endpoints on I. This is a less specific form of the above.

The first derivative test: If f ' x 0 exists and is positive, then f ' x is increasing at x 0.