There are
several different ways to use the calculator to find maximum or minimum
values of a function.
Of course,
the function has to have a formula, or the calculator can't process it
at all.
Suppose we
want to find the minimum value of x2 + 4x, where x
is any real number.
Regardless
of what technique we use to find a solution, it makes sense to graph
the function first, just to get an idea of what sort of answer to expect.
Graphical
From the
GRAPH screen, the Math menu (F5)
allows you to find a minimum value for a function you've graphed.
After selecting
Minimum, position the cursor to the left of the point you're interested
in and push ENTER.
Then move
the cursor to the right of the point you're interested in and push ENTER
again.
The calculator
returns the x-coordinate and the y-coordinate of the lowest point on the
graph.
Because
this is a graphical utility, it returns approximate
answers.
Automatic
On the HOME
screen, fMin is option 6 on the Calc
menu.
The calculator
returns the
x value which produces the smallest value for the function.
The calculator
computes an exact value if it can (assuming you're in auto
or exact mode).
These two
procedures work somewhat differently if you ask for a maximum.
Because
the graphical utility asks for a smallest and a largest value of x
to consider,
it will
return the largest value of the function between these two inputs.
fMax
doesn't do that.
If you want
to restrict the values of x that fMin or fMax consider,
use the with bar.
The command
for finding the maximum value of x2 + 4x for x
between -5 and 5 is fMax(x^2+4x,x)|x>=-5 and
x<=5.
Linear
Approximations
If we replace
x2 + 4x by a tangent line approximation, we're looking
for maximum or minimum values for a function whose graph is a straight
line.
This doesn't
work very well. (Why?)
In fact,
functions whose graphs are straight lines take on maximum and minimum values
only if the line is horizontal.
So we should
be looking for a tangent line approximation that's horizontal; that is,
we want an input where the derivative
of the function is zero.
zeros(d
(y1(x),x),x)
will return a list of the x-coordinates where the graph of y1 has
a horizontal tangent line.
Again, you
can use the with bar to restrict the range of x-values
the calculator will consider.
Exercises
Find the minimum and maximum values of x2 + 4x for x between -5 and 5.
Solve some of the maximum/minimum problems in the textbook using more than one of these techniques.