EXTREME VALUES OF FUNCTIONS Section 4 1 b

EXTREME VALUES OF FUNCTIONS Section 4. 1 b

Do Now Find the extreme values of: First, check the graph What does it suggest? What is the domain of the function? Since there are no endpoints, all extreme values must occur at critical points. Find the derivative:

Do Now Find the extreme values of: The only critical point in the domain is at x = 0… Check the original function: As x moves away from 0 on either side denominator gets smaller Value of f increases the graph rises We have a minimum!!! No Maxima, Minimum of 1/2 at x = 0

An Important Note Extrema can occur at critical points and endpoints, but not every critical point or endpoint automatically signals an extrema!!! Ex: Find the extrema of the following function using both analytic and graphical methods: Start with the graph… Derivative is never zero, and is undefined at x = 0.

An Important Note Extrema can occur at critical points and endpoints, but not every critical point or endpoint automatically signals an extrema!!! Ex: Find the extrema of the following function using both analytic and graphical methods: Start with the graph… Critical Point: Endpoint: But only the endpoint signals an extrema: Maximum of at

More Practice Problems Find the extreme values of the given function. Consider the graph! Critical points:

More Practice Problems Find the extreme values of the given function. Consider the graph! Critical points:

More Practice Problems Find the extreme values of the given function. Local Max of Local Min of at at

More Practice Problems Find the extreme values of the given function. First, sketch the graph… It appears that the derivative is zero at x = 0, and does not exist at x = 1. There appears to be a local max of 5 at x = 0 and a local min of 3 at x = 1.

More Practice Problems Find the extreme values of the given function. Critical Points at x = 0 and x = 1 How do we confirm this result analytically? Derivative is zero at x = 0 Left- and right-hand derivatives are not equal at x = 1 Local Max of 5 at x = 0, Local Min of 3 at x = 1

More Practice Problems Find the extreme values of the given function. Start by checking the graph… Derivative is zero at x = 0 and x = 12/5 Derivative is undefined at x = 3

More Practice Problems Find the extreme values of the given function. Start by checking the graph… C. P. Derivative Extremum Value x=0 0 Min. 0 x = 12/5 0 Local Max. x=3 Und. Min. 0