Definition of an Extrema
The Extreme Value Theorem
Definition of a Relative Extrema
Often we are considered how a point compares only with its
neighbors. If a function evaluated at a point is the largest among all
nearby function values, then we say that the function has a relative
maximum. Similarly, if the function evaluated at a point is
the largest among all nearby function values, then we say that the function
has a relative minimum.
Definition of a Critical Number
A value c is called a critical number of a function f if either
Now notice that since c is a relative maximum, the numerator is negative. Since the denominator takes on negative values for x < c and positive values for x > c, the derivative is both positive and negative. This can only occur if it is zero or does not exist.
From the two theorems, the extrema of a closed
interval can only occur at either a critical point or an end point. So to
find the extrema, set the derivative equal to 0, and solve. Plug the
solutions and the endpoints back into the original equation and the largest y
value will be the maximum, while the smallest will be the minimum.
Hence the maximum is 6 and occurs at x = 2, while the minimum is 2 and occurs at x = 1.