The First Derivative Test
The First Derivative Test
Recall that if f is a function, then f
has a relative maximum at x = c if for all points
a near c, f(c) >
f(a), and f has a relative minimum at x = c if for all
points a near c, f(c) <
f(a).
Consider a relative maximum, we have that on the left, the function is increasing
and on the right the function is decreasing. Similarly, for a relative
minimum, on the left the function
is decreasing and on the right the function is increasing.
We can now state the first derivative test:
The First Derivative Test
Let
f
be a differentiable function with
f '(c) = 0
then

If
f '(x)
changes from positive to negative, then
f
has a relative
maximum at
c.

If f
'(x) changes from negative to positive, then
f has a relative
minimum at c.

Example
Find and classify the relative extrema of
f(x) = x(1  x)^{2/5}
Solution
First set the first derivative equal to zero to locate the critical points.
f '(x) = (1  x)
^{2/5}  2/5 x(1  x) ^{3/5} =
0
Now multiply by (1  x) ^{3/5
} (1  x)  2/5 x = 0,
1  7/5 x = 0, x = 5/7 @ 0.714
So there is a critical point at 5/7. Notice
also that there is a critical point at x = 1 since the first
derivative is undefined there (notice the negative exponent 3/5). To determine whether the critical
point is a relative max, min or neither, choose a number just above and just
below the critical value.
x 
0.7 
0.8 
2 
f '(x) 
0.04 
0.3 
1.8 
Result 
Increasing 
Decreasing 
Increasing 
We see that f is increasing to the left of the critical number
and decreasing to the right of the critical number. Hence 5/7 is a
relative maximum.
The actual graph is shown below
Exercises

Classify the relative extrema of
f(x)= x + 1/x
Answer (Hold your mouse on the yellow rectangle)

An electric current in amps is given by
where w is a nonzero positive constant. Find the maximum values of the
current.
Answer (Hold your mouse on the yellow rectangle)
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