answersLogoWhite

0

Critical point (in one variable): A point on the graph y = f(x) at which f is differentiable and f'(x) = 0. The term is also used for the number c such that f'(c) = 0. The corresponding value f(c) is a critical value. A critical point c can be classified depending upon the behavior of fin the neighborhood of c, as one of following:

1. a local minimum, if f'(x) > 0 to the left of c and f'(x)< 0 to the right of c.

2. a local maximum, if f'(x) < 0 to the left of c and f'(x)> 0 to the right of c.

3. neither local maximum nor local minimum:

a) if f'(x) has the same sign to the left and to the right of c, in which case c is a horizontal point of inflection.

b) if there is an interval at every point of which f'(x) = 0 and c is an endpoint or interior point of this interval.

(This is called the first derivative test).

The second derivative test (is also a test for maximum and minimum values. It is a consequence of the concavity test):

Suppose f is continuous near c.

1. If f'(c) = 0 and f''(c) > 0, then f has a local minimum at c.

2. If f'(c) = 0 and f''(c) < 0, then f has a local maximum at c.

Example: f(x) = x^3 - 12x + 1

(a) Find the intervals on which f is increasing or decreasing.

(b) Find the local maximum and minimum values of f.

(c) Find the intervals of concavity and the inflection points.

Solution:

(a) f(x) = x^3 - 12x + 1

f'(x) = 3x^2 - 12

f'(x) = 3(x +2)(x - 2)

Interval: x < -2; -2 x < 2; x > 2

x + 2: - ; +; +

x - 2: - ; - ; +

f'(x): + ; - ; +

f: increasing on (-∞, -2); decreasing on (-2, 2); increasing on (2, ∞) So f is increasing on (-∞, -2) and (2, ∞) and f is decreasing on (-2, 2).

(b) f changes from increasing to decreasing at x = -2 and from decreasing to increasing at x = 2. Thus f(-2) = 17 is a local maximum value and f(2) = -15 is a local minimum value.

(c) f''(x) = 6x

f''(x) > 0 ↔ x > 0 and f''(x) < 0 ↔ x < 0. Thus f is concave upward on (0, ∞) and concave downward on (-∞, 0). There is an inflection point where the concavity changes, at (0, f(0)) = (0, 1).

User Avatar

Wiki User

16y ago

Still curious? Ask our experts.

Chat with our AI personalities

FranFran
I've made my fair share of mistakes, and if I can help you avoid a few, I'd sure like to try.
Chat with Fran
SteveSteve
Knowledge is a journey, you know? We'll get there.
Chat with Steve
JordanJordan
Looking for a career mentor? I've seen my fair share of shake-ups.
Chat with Jordan

Add your answer:

Earn +20 pts
Q: What is a critical point?
Write your answer...
Submit
Still have questions?
magnify glass
imp