For business calculus. I just want to check my answer ...
x is a member of [0,4] find the absolute extrema
To find the global extrema, check the local extrema and the limit of the function approaching the bounds.
Local extrema:
We have a local extremum at . What's f(x) of the point?
Now check the bounds.
We have 3 possible values to be the global extrema.
-- (the local extrema, x=1)
-- (left bound)
-- (right bound)
The greatest of these values, is the global maximum, and the least, is the global minimum.
The answer above is enough to solve. In addition, there are a few more things that may come in handy
Firstly, this is a quadratic (second degree) function.
Quadratic functions look like,
(position of the vertex and the functions closeness may vary)
Quadratic functions are of the form
The first coefficient indicates the side of the function (upwards or downwards) and its narrowness. If , the function is upwards. If , the function is downwards.
In this example, , which is greater than zero. This makes the function upwards. So, the vertex is the local and global minimum, and the global maximum is at the bounds.