4.1 - Maximum and Minimum Values
To manually find the maximum and minimum values of a function (without a calculator!), we have to use a combination of basic algebra and derivatives.
There are two types of maximums and minimums: Local and Absolute. Local maximums and minimums are extreme points on a graph that are only maximums in their own area or limited domain. Absolute maximums and minimums, on the other hand, are the very highest and lowest points on a graph (there may be more than one of each, but they would have to share the same Y value). A key indicator of an extreme point is a "kink" in the graph or a point at which the slope is 0.
To find a maximum, you must look at the critical points of a function and at the edges of the domain. For instance, on the graph of y=x^2 on the interval [-1,5], the minimum value would be (0,0) a critical point, while the maximum value would be (5,25), a point on the edge of the domain.
For example, if we were using the equation y= 3x^2 - 2x + 5 on [-2,4] and we wanted to find the extreme points in each direction, we would calculate the y values where the derivative is 0 and at the edges of the restricted domain. The edges of the domain are easy. We just plug -2 and 4 into the equation to get (-2,13) and (4,45). We now need to use the derivative equation to find the critical point(s).
The differentiated equation is y' = 6x-2. This means that at x = 1/3, the slope is horizontal. We now plug 1/3 back into the original equation to find the final critical point, (1/3, 14/3). Using this, we can now say that the maximum in the domain is (4,45) and the minimum is (1/3, 14/3).
Many times, you will also be required to find the local maximums and minimums. They are also critical points and "kinks", but they are not always the absolute minimums or maximums. However, any critical point or "kink" at which the sign (positive or negative) of the slope is different on both sides is a local minimum or maximum.
Here's an interesting website that explains maximums and minimums in more depth.
Lauren's next....But she doesn't have to post until thursday...
And so I leave you with yet another beloved Grandpa Simpson quote...
"My Homer is not a communist. He may be a liar, a pig, an idiot, a communist, but he is not a porn star. "