What is the minimum of this graph?

[ikilemon]

Question: Where and what value is the minimum?
I got confused due to the open circle on the graph, my answer for the maximum was:
The max value is at x=2 with a value of 4, if that helps understand the question better

 

[lev888]

Please post the definition of the minimum of a function.

 

[HallsofIvy]

Are you trying to find local minimums or the global minimum?

A "local" minimum is a point where the function value is less than or equal to the values at every point in some neighborhood of the point.

A "global" minimum is a point where the function value is less than or equal to the value at every point.

There is a local minimum of y= 0 at x= -3. There is NO global minimum. y gets arbitrarily close to -2 as x approaches 0 from the left but does not actually take on that value.

 

[Steven G]

Think about it this way, how can the minimum be at y=-2 if no point on that graph ever equal -2?
The answer would have been -2 but there is that open circle. For the same reason y=-5 is not the min (why?), y=-2 is not the min.

 

[HallsofIvy]

You might want to think about the difference between a "minimum" and a "greatest lower bound". A minimum for a set of values (such as the values of a function) is a number that is in the set, and is less than or equal to any number in the set. A "lower bound" is a number, not necessarily in the set, that is lower than or equal to any number in the set. For example, for the set (0, 1), the set of all real numbers that are larger than 0 and less than 1, does not have a "minimum", since if "x" is any number in that set , x/2 is also in the set and is smaller. The greatest lower bound is 0. The set of "lower bounds" contains all negative numbers and 0, The largest of those is 0.

Every set of real numbers, having a lower bound, has a greatest lower bound but not necessarily a minimum. Of course, if a set has a minimum, then that minimum is also the greatest lower bound.

For a function having this graph, there is no "global minimum" but the greatest lower bound for the set of y values is -2.