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A point c in the domain of a function f is called a critical point of f if
- a)f’ (x) = 0 at x = c
- b)f is not differentiable at x = c
- c)Either f’ (c) = 0 or f is not differentiable
- d)f” (x) = 0, at x = c
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A point c in the domain of a function f is called a critical point of ...
A point C in the domain of a function f at which either f(c) = 0 or f is not differentiable.
The point f is called the critical point.
c is called the point of local maxima
If f ′(x) changes sign from positive to negative as x increases through c, that is, if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c.
c is called the point of local minima
If f ′(x) changes sign from negative to positive as x increases through c, that is, if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c.
c is called the point of inflexion
If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima.
The point f is called the critical point.
c is called the point of local maxima
If f ′(x) changes sign from positive to negative as x increases through c, that is, if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c.
c is called the point of local minima
If f ′(x) changes sign from negative to positive as x increases through c, that is, if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c.
c is called the point of inflexion
If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima.
Most Upvoted Answer
A point c in the domain of a function f is called a critical point of ...
Answer:
To understand why option 'B' is the correct answer, let's break down the definitions and concepts involved.
Definition of a Critical Point:
A point c in the domain of a function f is called a critical point of f if the function is not differentiable at x = c.
What is Differentiability?
Differentiability is a property of a function that indicates whether the function has a derivative at a particular point. A function is said to be differentiable at a point if the derivative exists at that point.
Explanation:
Option 'B' states that the function is not differentiable at x = c. This means that the derivative of the function does not exist at x = c.
Why is this a critical point?
In order for a function to have a derivative at a point, the function must be continuous at that point. If a function is not continuous at a point, it cannot have a derivative at that point. Therefore, when a function is not differentiable at a point, it implies that the function is not continuous at that point.
Connection to f(x) = 0:
The fact that f(x) = 0 at x = c is not relevant to the definition of a critical point. A function can have critical points even if f(x) ≠ 0 at x = c. Therefore, option 'D' (f(x) = 0 at x = c) does not make a point critical.
Conclusion:
Based on the definitions and concepts discussed, the correct option is 'B' because a critical point is defined as a point where the function is not differentiable. The condition f(x) = 0 at x = c is not necessary for a point to be considered critical.
To understand why option 'B' is the correct answer, let's break down the definitions and concepts involved.
Definition of a Critical Point:
A point c in the domain of a function f is called a critical point of f if the function is not differentiable at x = c.
What is Differentiability?
Differentiability is a property of a function that indicates whether the function has a derivative at a particular point. A function is said to be differentiable at a point if the derivative exists at that point.
Explanation:
Option 'B' states that the function is not differentiable at x = c. This means that the derivative of the function does not exist at x = c.
Why is this a critical point?
In order for a function to have a derivative at a point, the function must be continuous at that point. If a function is not continuous at a point, it cannot have a derivative at that point. Therefore, when a function is not differentiable at a point, it implies that the function is not continuous at that point.
Connection to f(x) = 0:
The fact that f(x) = 0 at x = c is not relevant to the definition of a critical point. A function can have critical points even if f(x) ≠ 0 at x = c. Therefore, option 'D' (f(x) = 0 at x = c) does not make a point critical.
Conclusion:
Based on the definitions and concepts discussed, the correct option is 'B' because a critical point is defined as a point where the function is not differentiable. The condition f(x) = 0 at x = c is not necessary for a point to be considered critical.
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Community Answer
A point c in the domain of a function f is called a critical point of ...
Critical point
f'(c)=0
or f is not differentiable at c
f'(c)=0
or f is not differentiable at c
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A point c in the domain of a function f is called a critical point of f ifa)f’ (x) = 0 at x = cb)f is not differentiable at x = cc)Either f’ (c) = 0 or f is not differentiabled)f” (x) = 0, at x = cCorrect answer is option 'B'. Can you explain this answer?
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