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When you're searching for the high and low points of a function's graph, you're looking for what's called the critical points. A critical point occurs where the function's first derivative is either zero or undefined. This indicates a potential maximum, minimum, or point of inflection.

In mathematical terms, if you have a function denoted as f(x), you find its critical points by taking the derivative, f'(x), and setting it to zero to solve for x. If f'(x) doesn't exist at certain x-values within the domain, those are critical points as well. But remember, not all critical points correspond to an extremum, which is why further analysis, like the first derivative test, is usually required.

For instance, with the function given as f(x) = cos(蟺x), you would calculate its derivative as f'(x) = -蟺 sin(蟺x). In our case, since the derivative never equals zero on the specified interval and is never undefined, we find no critical points inside the interval. This lack of internal critical points hints that the extrema might be located at the boundary points of the interval.

The First Derivative Test is essentially a 'detective tool' for finding local maxima and minima of a function based on its first derivative. After finding the critical points of a function, this test allows you to determine whether each point is a local maximum, local minimum, or neither.

How does it work? You derive the function and analyze the sign of the derivative before and after each critical point. If the derivative changes from positive to negative at a point, it's a local maximum. Conversely, if it changes from negative to positive at the critical point, we have a local minimum. If there's no change in the sign, then the point is not an extremum.

However, for the function f(x) = cos(蟺x) on the interval [0, 1/6], since there were no critical points within the interval, the First Derivative Test doesn't provide additional information regarding the extrema within the interval. Yet, it remains an important concept for various functions with critical points within the domain of interest.

Sometimes the extrema of a function occur not in the middle of the graph but at the 'ends'鈥攁nd that鈥檚 what Closed Interval Analysis is all about. This method involves analyzing functions over closed intervals [a, b], which include their endpoint values.

Here's how you do it:

• Find all critical points within the interval.
• Evaluate the function at each critical point.
• Evaluate the function at the endpoint values of the interval.
• Compare all these values to determine the absolute (global) maximum and minimum.

For the cos(蟺x) function on the interval [0, 1/6], we check the function's value at endpoints because no internal critical points were found. At the left endpoint, x=0, the function has a value of 1, and at the right endpoint, x=1/6, the value is 鈭�3/2. So, in this case, applying closed interval analysis concludes that the function reaches its absolute maximum at x=0 and absolute minimum at x=1/6.