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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).
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What is the critical level of a backflow preventer?
the critical level of a backflow preventer is the point at which the outgoing pressure exceeds the incoming pressure
Is critical point also an inflection point?
no, a critial point is where the slope (or the derivitive) is 0. the inflection point is when the graph switches from concave up to concave down or vice versa
How do you calculate critical points of derivatives?
The "critical points" of a function are the points at which the derivative equals zero or the derivative is undefined. To find the critical points, you first find the derivative of the function. You then set that derivative equal to zero. Any values at which the derivative equals zero are "critical points". You then determine if the derivative is ever undefined at a point (for example, because the denominator of a fraction is equal to zero at that point). Any such points are also called "critical points". In essence, the critical points are the relative minima or maxima of a function (where the graph of the function reverses direction) and can be easily determined by visually examining the graph.
What is important to remember about critical points that are zeros of the denominator?
If the denominator is zero at some point, then the function is not defined at the corresponding points.
