Calculus I
Limits
Derivative
Exercices
Applications
Marginal analysis
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Calculus I: Derivative
First Derivative Test
1. The First Derivative Test
Suppose f is continuous at a critical point xo.
• If f'(x) > 0 on an open interval extending left from xo and f'(x) < 0 on an open interval extending right from xo, then f has a relative maximum at xo.
• If f'(x) < 0 on an open interval extending left from xo and f'(x) > 0 on an open interval extending right from xo, then f has a relative minimum at xo.
• If f'(x) has the same sign on both an open interval extending left from xo and an open interval extending right from x0, then f does not have a relative extremum at xo.
• In summary, relative extrema occur where f(x) changes sign.
2. Example
Let's consider, on the interval [ 2, 3], the following function:
f(x) = 3 x^{4}  4 x^{3} 12 x^{2} + 3
This function is differentiable everywhere on the interval [ 2, 3].
Taking the derivative, we obtain:
f'(x) = 12 x^{3}  12 x^{2} 24 x =
12 x ( x^{2}  x  2) =
12 x (x  2)(x + 1)
f(x) = 0 for x =  1, 0, and + 2.
These are the three critical points of f on [ 2, 3].
x   ∞ 
 1  0  2 
+ ∞ 
x + 1   
  +  + 
+ 
x   
    + 
+ 
x  2   
     
+ 
f(x)   
+   
+ 
+ 
By the First Derivative Test, f has a relative maximum at
x = 0 and relative minima at x =  1 and x = 2.

