Problem
9: 1998-1999
The set of graphs below are f(x), f '(x), and f
"(x), but not necessarily in that order.
(a) Determine which graph represents
which function.
(b) Describe and draw possible graphs of f '''(x) and
f(4)(x), the third and fourth derivatives.
Solution:
(a) The function is blue, its first derivative is red, and the
second derivative is green. Notice that whenever the blue graph has a relative
max or min, the red graph has a value of zero (since the tangent line to the
function will have a slope of zero whenever the function has a max or min).
Also, whenever the the blue graph is increasing, the red graph is positive (above
the x-axis), and similarly, when the blue graph is decreasing, the red
graph is negative (below the x-axis).
Notice that the green graph acts in a similar fashion
to the red graph. When the red graph hits its only relative extrema point (a
max), the green graph has a value of zero. In addition, the sign of the green
graph describes the direction of the red graph. Lastly, the green graph is positive
whenever the blue is concave up and is negative whenever the blue is concave
down. (The green hits the x-axis at the red max and the blue inflection
point.)
It is not
good enough to say that the blue function might be a fifth-degree polynomial,
the red a fourth, and the green a third. This is conjecture and is not
strong support for your answer. It's a good observation, but not sufficient
support.
(b) The third derivative must describe
the direction of the green graph (note that it must always be negative
since the green is always decreasing). Also, the third derivative must
describe the concavity of the red (which is always down, and therefore
always negative -- these thoughts seem to mesh well indeed.) Similarly,
the fourth derivative describes the direction of the third derivative and
the concavity of the second derivative (green). Their graphs are shown
below. The third derivative is orange and the fourth is purple. (By the
way, the blue was a fifth degree polynomial and the purple is clearly linear.
Not all graphs, however, will simplify down to a line as you take derivatives.
Think about cosine's derivatives, for example.)
