8.2.4 The First Derivative Test

Question 1

The First Derivative Test

Answer 1

• Determining the sign of the derivative immediately to either side of a critical point reveals whether that point is a relative maximum, relative minimum, or neither.
  • The first derivative test states that if c is a critical point of f, and f is continuous and differentiable on an open interval containing c (except possibly at c), then f(c) can be classified as follows:
  1) If f (c) < 0 for x < c and f (c) > 0 for x > c, then f(c) is a relative minimum of f.
  2) If f (c) > 0 for x < c and f (c) < 0 for x > c, then f(c) is a relative maximum of f.
  • On an interval, the sign of the first derivative of a function indicates whether that function is increasing or decreasing.

Question 2

note

Answer 2

• Maximum and minimum values of a function will only occur at critical points: places where the derivative is zero or undefined.
• Once you have identified the critical points, use a sign chart to illustrate the behavior of the graph on either side of each critical point.
• The sign chart for g (x) shows that the function g is
  increasing to the left of x = –1 and decreasing to the right. Therefore, g has a maximum value at x = –1. Since the g is decreasing to the left of x = 3 and increasing to the right of it, there is a minimum value at x = 3.
• Take a look at the possible situations around a critical point.
• If the function changes from increasing to decreasing, it will have a relative maximum value at the critical point. If the function changes from decreasing to increasing, it will have a relative minimum at the critical point.
• If the function does not change behavior on either side of the critical point, it will have neither a maximum value nor a minimum value at the critical point.
• Notice the general relationship between the sign of the
  derivative, the regions where the function is increasing or decreasing, and the existence of maximum or minimum points. Using the first derivative test, you can move directly from the sign of the derivative to the existence of maxima and minima.

Question 3

Suppose f (x) is continuous and defined for all real numbers. You are given that f (x) has critical points at x = −1 and x = 0. If f ′(x) is positive in the interval x < −1, negative in the interval −1 < x < 0, and positive in the interval x > 0, is the point where x = 0 the location of a relative maximum, minimum, or neither?

Answer 3

Minimum

Question 4

Suppose g(x)is continuous and defined for all real numbers. You are given that g(x)has critical points at x=−5/2andx=−1. If g’(x)is positive in the interval x−1,is the point where x=−5/2 the location of a relative maximum, minimum, or neither?

Answer 4

Neither

Question 5

Suppose that h (x) is a continuous function and is defined for all x greater than or equal to 1. You are given that h (x) has critical points at x = 1, x = 3, and x = 5. If h′ (x) is negative on the interval 1 < x < 3, positive on the interval 3 < x < 5, and positive on the interval x > 5, what can be said about the point (3, h (3))?

Answer 5

(3, h (3)) is an absolute minimum