Projects - Cengage Learning

C-32 Appendix C
Setting 2x 4 x, we obtain 3x 4, and consequently, x 43. This is
the required x-coordinate of the turning point. For the y-coordinate we have
4 4
512
f 1 32 21 3214 4 32 2 . As you can verify, that works out to In summary, the
4
coordinates of the turning point are 1 3, 512
27 .
27 2.
Third sample problem: According to Figure B, the graph of the polynomial
function defined by g(x) x 2 (1 x) has a turning point in the first quadrant.
Find the x-coordinate of that turning point. First, note that on the open interval
0 x 1, each of the three factors x, x again, and 1 x is positive. However,
as you can check, the sum of the three factors is not constant. We can work
around this by writing
x
(x)(x)(1 x) 4 ca
2 ba x
b (1 x)d
2
Graphical Perspective
1
0
Figure B
g(x) x 2 (1 x)
_1
_1 0 1 2
Now notice that the three factors in brackets do have a constant sum. For a
x
maximum then, we require that 2 1 x. Solving this last equation yields
x 23, the required x-coordinate.
Exercises
1. (a) First, sketch a graph of the function f(x) x 2 (6 2x) using the standard
techniques from Section 4.6. Then, use the ideas presented above
to find the exact coordinates of the turning point in the first quadrant.
(b) Follow part (a), but use g(x) x 2 (6 3x).
2. An open-top box is to be constructed from a 16-inch square sheet of cardboard
by cutting out congruent squares at each corner and then folding up the
flaps. See the following figure. Express the volume of the box as a function
of a single variable. Then find the maximum possible volume for the box.
16 in.
x
x
16 in.
x