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

1. Classify the relative extrema of

f(x)= x + 1/x 2. An electric current in amps is given by where w is a nonzero positive constant.   Find the maximum values of the current. 