answersLogoWhite

0


Best Answer

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

15y ago
This answer is:
User Avatar

Add your answer:

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

What does the critical point represent?

Critical point is also known as a critical state, occurs under conditions at which no phase boundaries exist. There are multiple types of critical points, including vapor-liquid critical points and liqui-liquid critical points.


How does density vary for a component at its critical point?

For a pure component the density difference between a liquid and gas approaches zero as the critical point is approached. By definition liquid and gaseous phases are indistinguishable (meaning no difference) at the critical point.


What is the critical temperature and pressure of CCl4?

The critical temperature of CCl4 (carbon tetrachloride) is 283.5°C and the critical pressure is 45.6 atm. At the critical point, the distinction between liquid and gas phases of CCl4 disappears, and it behaves as a supercritical fluid.


What is a critical control point when preparing a beef curry?

Which of the following is a critical control point when preparing a beef curry?


What is a substance's critical point?

A substance's critical point is the temperature and pressure at which the gas and liquid phases of the substance become indistinguishable, forming a supercritical fluid. At the critical point, the substance exhibits unique properties, such as density and viscosity, that differ from those of its gas or liquid phases.


What is Critical point of phenol-water system?

The critical point of the phenol-water system occurs when the temperature and pressure reach the critical values at which the distinction between liquid and gas phases disappears. At this point, the properties of both phases become indistinguishable, leading to a critical point that represents the maximum temperature and pressure at which the two phases can coexist.


The rejection and non rejection regions are divided by the?

The rejection and non rejection regions are divided dividing point. critical value. point of no return. rejection value


What is the critical point called?

The critical point is called the point at which a function's derivative is zero or undefined. At this point, the function may have a local maximum, minimum, or an inflection point.


What are critical and super critical boilers?

critical boiler operating steam parameter 221.12bar and 375 centigrate . you can rise in pressure and temperatureabove the critical point. that is called supercritical boiler.


What is the point where the gas and liquid phase become indistinguishable from each other?

This point is called critical point.


What is the value of dryness fraction at critical point?

at critical ponint dryness factor is either 0 or 1..


The density of a gas is maximum at?

It's Critical Point?