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Find the critical numbers and stationary points of each of the following functions.$\;f(x)=x^{3}-3x+1$

Note: This is part 2nd of a 6 part question, split as 6 separate questions here.

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Toolbox:
  • A ciritical number of a function f is a number c in the domain of f such that either $f'(c)=0$ or $f'(c)$ does not exist.
  • Stationary points correspond to critical numbers for which $f'(c)=0$
$f(x)=x^3-3x+1$
Step 1:
$f'(x)=3x^2-3$
$\qquad=3(x^2-3)$
$\qquad=3(x^2-1)$
$\qquad=3(x+1)(x-1)$
$f'(x)=0=>3(x+1)(x-1)=0=>x=-1,1$ (critical numbers)
Step 2:
When $x=-1,f(-1)=-1+3+1=3$
When $x=1, f(1)=1-3+1=-1$
The critical points are $(-1,3)$ and $(1,-1)$

 

answered Jul 31, 2013 by meena.p
edited Jul 31, 2013 by meena.p
 

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