1. Graph of a Function
   To sketch the graph of a differentiable function , follow these steps:
1. Find and determine the values of where .
2. Check the signs of on both sides of these -values and create an increasing/decreasing chart (sign chart).
3. Use the information about the function’s increasing/decreasing behavior, local maximum/minimum points, and intercepts with the coordinate axes to sketch the general shape of the graph.


2. Maximum and Minimum Values of a Function
   When a function is continuous on a closed interval , its maximum and minimum values can be found using the following steps:
1. Find the local maximum and minimum values of in the open interval .
2. Evaluate the function at the endpoints and .
3. The largest of these values is the maximum, and the smallest is the minimum.


3.  
1. i.Local maximum and minimum values are not always the global maximum and minimum.
2. ii.If is continuous on a closed interval, there will always be exactly one global maximum and one global minimum, though there may be several local extrema.
3. iii.If a continuous function has only one extreme value (either a maximum or a minimum) on the interval :
4. If the extreme value is a local maximum, then
   (local maximum) (global maximum)
5. If the extreme value is a local minimum, then
   (local minimum) (global minimum)


4. Application of Maximum and Minimum Values of a Function
   To find the maximum or minimum value of quantities like length, area, or volume:
1. Define an appropriate variable as the unknown .
2. Express the quantity to be maximized or minimized as a function of .
3. Differentiate the function and find the critical points (extrema).
4. Ensure that the maximum or minimum value is within the allowable range for , then determine the desired maximum or minimum value.