Minimum and Maximum Values of Functions 4.6 Minimum and Maximum Values of Functions Borax Mine, Boron, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington

Extreme values can be in the interior or the end points of a function. No Absolute Maximum Absolute Minimum

Absolute Maximum Absolute Minimum

Absolute Maximum No Minimum

No Maximum No Minimum

Extreme Value Theorem: If f is continuous over a closed interval, then f has a maximum and minimum value over that interval. Maximum & minimum at interior points Maximum & minimum at endpoints Maximum at interior point, minimum at endpoint

Finding Maximums and Minimums Analytically: 1 Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points. 2 Find the value of the function at each critical point. 3 For closed intervals, check the end points as well. For open intervals, check the limits as x approaches the end points to determine if min or max exist within interval.

EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on the interval . There are no values of x that will make the first derivative equal to zero. The first derivative is undefined at x=0, so (0,0) is a critical point. Because the function is defined over a closed interval, we also must check the endpoints.

At: At:

At: At: At: Absolute minimum: maximum:

EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on interval . Critical point at x = 1.5. Because the function is defined over a open interval, we also must check the limits at the endpoints.

Absolute minimum: maximum: