Simple Problems on Maxima/Minima A maximum is the point in a function's domain that corresponds to the highest point or peak. A minimum is the point...
A maximum is the point in a function's domain that corresponds to the highest point or peak. A minimum is the point in the domain that corresponds to the lowest point or valley.
To find the maximum or minimum of a function, we look for the point(s) where the first derivative is equal to zero. These points are called critical points.
Critical points are classified into three categories:
Local maximums: occur when the first derivative is positive.
Local minima: occur when the first derivative is negative.
Saddle points: occur when the first derivative is zero but not positive or negative.
Identifying critical points is important because they determine the turning points of the function. A function can have at most one local maximum, one local minimum, and one saddle point.
In addition to finding critical points, we can also find the maximum or minimum by evaluating the function at the critical points.
Examples:
Therefore, 0 is the only critical point. Evaluating the function at this point gives us a maximum of 2.
Therefore, 1 is the only critical point. Evaluating the function at this point gives us a minimum of 0.
Tips for Finding Maxima/Minima:
Look for points where the first derivative is equal to zero.
Use a graphing calculator or derivative calculator to find critical points.
Identify the category of each critical point by looking at the sign of the first derivative.
Evaluate the function at the critical point to find the maximum or minimum value