Page 100 - Calculus Demystified

CHAPTER 3 Applications of the Derivative
at a local maximum. This is sometimes called Fermat’s test. Also, we see that the 87
graph is concave down at a local maximum.
It is common to refer to the points where the derivative vanishes as critical points.
In some contexts, we will designate the endpoints of the domain of our function to
be critical points as well.
Now look at a local minimum. Notice that a minimum has the characterizing
property that it looks like a valley: the function is decreasing to the left of the valley
and increasing to the right of the valley. The derivative at the valley is 0: the function
neither increases nor decreases at a local minimum. This is another manifestation
of Fermat’s test. Also, we see that the graph is concave up at a local minimum.
Let us now apply these mathematical ideas to some concrete examples.
EXAMPLE 3.6
3
2
Find all local maxima and minima of the function k(x) = x −3x −24x +5.
SOLUTION
We begin by calculating the first derivative:
2

k (x) = 3x − 6x − 24 = 3(x + 2)(x − 4).

We notice that k vanishes only when x =−2or x = 4. These are the only

candidates for local maxima or minima. The second derivative is k (x) =

6x − 6. Now k (4) = 18 > 0, so x = 4 is the location of a local minimum.
Also k (−2) =−18 < 0, so x =−2 is the location of a local maximum.

A glance at the graph of this function, as depicted in Fig. 3.10, confirms our
calculations.
EXAMPLE 3.7
Find all local maxima and minima of the function g(x) = x + sin x.

SOLUTION
First we calculate that

g (x) = 1 + cos x.
Thus g vanishes at the points (2k + 1)π for k = ... , −2, −1, 0, 1, 2,....

Now g (x) = sin x. And g ((2k + 1)π) = 0. Thus the second derivative test
is inconclusive. Let us instead look at the first derivative. We notice that it is
always ≥ 0. But, as we have already noticed, the first derivative changes sign at
a local maximum or minimum. We conclude that none of the points (2k + 1)π
is either a maximum nor a minimum. The graph in Fig. 3.11 confirms this
calculation.
3
You Try It: Find all local maxima and minima of the function g(x) = 2x −
2
15x + 24x + 6.