Expert: Paul Klarreich - 3/2/2011

Question
Hi Paul,

The following if my question:
f(x)= (x^2/3)/1+x+x^4
I am supposed to use a computer system to graph f, f prime, and f". I am then supposed to use the graphs to estimate the intervals of increase, decrease, extreme values, intervals of concavity, and inflection points of f.
I know how to find the different equations of f, f prime, and f", and I know how to graph. However, how am I supposed to figure out the rest of the problem?

Answer
Questioner: Alyssa Zima
Country: United States
Category: Calculus
Private: No
Subject: Calculus - graphing with calculus and calculators
Question: Hi Paul,

The following if my question:
f(x)= (x^2/3)/1+x+x^4
I am supposed to use a computer system to graph f, f prime, and f". I am then supposed to use the graphs to estimate the intervals of increase, decrease, extreme values, intervals of concavity, and inflection points of f.
I know how to find the different equations of f, f prime, and f", and I know how to graph. However, how am I supposed to figure out the rest of the problem?
................................................
Analytic geometry pairs:

GEOMETRIC observation/fact <----> ALGEBRAIC observation/fact

GEOM.                                 ALG.
--------------------------------------------------------------------------------
Graph goes down                       f' is neg.
Graph goes up                         f' is pos.
Horizontal tangent                    f' = 0
Graph goes down,then up.              f has a rel min.
Graph goes up,then down.              f has a rel max.
Graph turns left(conc up)             f'' > 0
Graph turns right(conc dn)            f'' < 0
.. stops turning left,starts          f'' = 0
     turning right.(or v-v)

And there are a few more.  But here is the basic thing to know and accept about graph sketching:

IT TAKES TIME!  Each example will take 30 minutes.  Try to do it too fast and you get stuck.

Analyze:

Asymptotes -- vertical, horizontal, oblique.
Intercepts -- x- and y-.
First derivative tests: rising, falling, critical pts, min, max.
Second ...: concave up (turning left) vs concave down.  Infl. pts.
First AND second: rel min,max.

Your book tells you how to do all these.