Expert: Paul Klarreich - 12/8/2007

Question
Hi, I have to graph the equation f(x)=x+ (32/x^2) and I am confused because I do not know how to find the critical numbers to begin the problem when there is only part of the equation in a fraction. I know that I have to do the first and 2nd derivative tests, but what should I do to find the critical number?

Answer
Questioner:   Annie
Category:  Calculus
Private:  No
 
Subject:  Finding critical #
Question:  Hi, I have to graph the equation f(x)=x+ (32/x^2) and I am confused because I do not know how to find the critical numbers to begin the problem when there is only part of the equation in a fraction. I know that I have to do the first and 2nd derivative tests, but what should I do to find the critical number?
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Hi, Annie,

You find the critical numbers the same way, whether there is one fractional part or 50.

Critical numbers are numbers:
1. Where f'(x) = 0 [These are called Stationary points]
2. Where f'(x) is undefined. [These are called Singular points]
3. End points of the interval, if any. [These are called, er.. End points.]


You have f(x) = x - 32x^-2   << that's the way to write it for this purpose.

f'(x) = 1 + 64x^-3
            64
f'(x) = 1 + -----
            x^3

Set that equal to zero:

x^3 = - 64,   x = -4 is a staionary point.
Also, x = 0 is a singular point, because f'(0) is undefined.

Now:

f'(x) = 1 - 64x^-3

f''(x) = 192x^-4 = 192/x^4.

There is no inflection point.