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:
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Let f be a differentiable function with f '(c) = 0 then
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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
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Classify the relative extrema of
f(x)= x + 1/x
Answer (Hold your mouse on the yellow rectangle)
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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)