when .
Sign plot of or graph of : min. , max. and min. .
The x-intersects of are .
The x-intersects of are and .
You do not need to differentiate : the graph is an open down parabola with max. .
For the following is true: when .
The extrema of are therefore: max. and min. .
Sports field: length m and width m where is the radius of each of the two half circles.
The following must be true: , and therefore .
The surface area of the sports field is: .
Calculating te maximum: gives you .
The sports field therefore has a length of m and a width of m.
euro per kg.
Use your graphing calculator to make a table of and compare the values to those in the table.
is the speed with which the profit changes as increases.
when .
The maximum profit is about .
gives you .
This means that the function will only have a minimum other than when . In that case the minimum is .
The minimum is equal to if and therefore if ( is not allowed).
and , so the tangent must have equation .
At the same time, must be a solution of this equation: gives you .
gives you .
When the graph of always has two extrema.
gives you and therefore and .
This is a minimum.