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Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?
- a)It is continuous on its domain.
- b)It has a local minimum at x = 2.
- c)It has a local maximum at x = 2.
- d)It is differentiable on its domain.
Correct answer is option 'A,B'. Can you explain this answer?
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Let max {a, b} denote the maximum of two real numbers a and b. Which o...
Statement: Let max {a, b} denote the maximum of two real numbers a and b.
Given function: f(x) = max{3 - x, x - 1}
Domain: The domain of the function is the set of all real numbers.
Statement a): It is continuous on its domain.
To determine if the function is continuous on its domain, we need to check if it is continuous at every point within its domain.
- For x < 1,="" the="" function="" evaluates="" to="" f(x)="max{3" -="" x,="" x="" -="" 1}="max(3" -="" x,="" x="" -="" 1)="3" -="" x.="" />
- The function is continuous for x < 1="" since="" it="" is="" a="" linear="" />
- For 1 ≤ x ≤ 3, the function evaluates to f(x) = max{3 - x, x - 1} = max(3 - x, x - 1) = x - 1.
- The function is continuous for 1 ≤ x ≤ 3 since it is a linear function.
- For x > 3, the function evaluates to f(x) = max{3 - x, x - 1} = max(3 - x, x - 1) = x - 1.
- The function is continuous for x > 3 since it is a linear function.
Since the function is continuous at every point within its domain, statement a) is true.
Statement b): It has a local minimum at x = 2.
To find the local minimum of the function, we need to analyze the behavior of the function around x = 2.
- For x < 2,="" the="" function="" evaluates="" to="" f(x)="max{3" -="" x,="" x="" -="" 1}="max(3" -="" x,="" x="" -="" 1)="3" -="" />
- As x approaches 2 from the left, f(x) approaches 3 - 2 = 1.
- Therefore, there is no local minimum at x = 2 from the left.
- For x > 2, the function evaluates to f(x) = max{3 - x, x - 1} = max(3 - x, x - 1) = x - 1.
- As x approaches 2 from the right, f(x) approaches 2 - 1 = 1.
- Therefore, there is no local minimum at x = 2 from the right.
Since the function does not have a local minimum at x = 2, statement b) is false.
Statement c): It has a local maximum at x = 2.
Since the function does not have a local minimum at x = 2, it also does not have a local maximum at x = 2. Therefore, statement c) is false.
Statement d): It is differentiable on its domain.
To determine if the function is differentiable on its domain, we need to check if it is differentiable at every point within its domain.
- For x < 1,="" the="" function="" evaluates="" to="" f(x)="max{3" -="" x,="" x="" -="" 1}="max(3" -="" x,="" x="" -="" 1)="3" -="" />
- The function is differentiable
Given function: f(x) = max{3 - x, x - 1}
Domain: The domain of the function is the set of all real numbers.
Statement a): It is continuous on its domain.
To determine if the function is continuous on its domain, we need to check if it is continuous at every point within its domain.
- For x < 1,="" the="" function="" evaluates="" to="" f(x)="max{3" -="" x,="" x="" -="" 1}="max(3" -="" x,="" x="" -="" 1)="3" -="" x.="" />
- The function is continuous for x < 1="" since="" it="" is="" a="" linear="" />
- For 1 ≤ x ≤ 3, the function evaluates to f(x) = max{3 - x, x - 1} = max(3 - x, x - 1) = x - 1.
- The function is continuous for 1 ≤ x ≤ 3 since it is a linear function.
- For x > 3, the function evaluates to f(x) = max{3 - x, x - 1} = max(3 - x, x - 1) = x - 1.
- The function is continuous for x > 3 since it is a linear function.
Since the function is continuous at every point within its domain, statement a) is true.
Statement b): It has a local minimum at x = 2.
To find the local minimum of the function, we need to analyze the behavior of the function around x = 2.
- For x < 2,="" the="" function="" evaluates="" to="" f(x)="max{3" -="" x,="" x="" -="" 1}="max(3" -="" x,="" x="" -="" 1)="3" -="" />
- As x approaches 2 from the left, f(x) approaches 3 - 2 = 1.
- Therefore, there is no local minimum at x = 2 from the left.
- For x > 2, the function evaluates to f(x) = max{3 - x, x - 1} = max(3 - x, x - 1) = x - 1.
- As x approaches 2 from the right, f(x) approaches 2 - 1 = 1.
- Therefore, there is no local minimum at x = 2 from the right.
Since the function does not have a local minimum at x = 2, statement b) is false.
Statement c): It has a local maximum at x = 2.
Since the function does not have a local minimum at x = 2, it also does not have a local maximum at x = 2. Therefore, statement c) is false.
Statement d): It is differentiable on its domain.
To determine if the function is differentiable on its domain, we need to check if it is differentiable at every point within its domain.
- For x < 1,="" the="" function="" evaluates="" to="" f(x)="max{3" -="" x,="" x="" -="" 1}="max(3" -="" x,="" x="" -="" 1)="3" -="" />
- The function is differentiable
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Community Answer
Let max {a, b} denote the maximum of two real numbers a and b. Which o...
For finding intersection point 3 - x = x - 1
At x = 2
So, 3 - x and x - 1 intersects at x = 2.
Check for continuity
f(2-) = f(2+) = f(2)
It is continuous in its domain.
Check for differentiability
It is not differentiable.
For maximum and minima
It has local minima at x = 2
At x = 2
So, 3 - x and x - 1 intersects at x = 2.
Check for continuity
f(2-) = f(2+) = f(2)
It is continuous in its domain.
Check for differentiability
It is not differentiable.
For maximum and minima
It has local minima at x = 2
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Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?a)It is continuous on its domain.b)It has a local minimum at x = 2.c)It has a local maximum at x = 2.d)It is differentiable on its domain.Correct answer is option 'A,B'. Can you explain this answer?
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Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?a)It is continuous on its domain.b)It has a local minimum at x = 2.c)It has a local maximum at x = 2.d)It is differentiable on its domain.Correct answer is option 'A,B'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?a)It is continuous on its domain.b)It has a local minimum at x = 2.c)It has a local maximum at x = 2.d)It is differentiable on its domain.Correct answer is option 'A,B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?a)It is continuous on its domain.b)It has a local minimum at x = 2.c)It has a local maximum at x = 2.d)It is differentiable on its domain.Correct answer is option 'A,B'. Can you explain this answer?.
Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?a)It is continuous on its domain.b)It has a local minimum at x = 2.c)It has a local maximum at x = 2.d)It is differentiable on its domain.Correct answer is option 'A,B'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?a)It is continuous on its domain.b)It has a local minimum at x = 2.c)It has a local maximum at x = 2.d)It is differentiable on its domain.Correct answer is option 'A,B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f{x) = max{3 - x, x - 1}?a)It is continuous on its domain.b)It has a local minimum at x = 2.c)It has a local maximum at x = 2.d)It is differentiable on its domain.Correct answer is option 'A,B'. Can you explain this answer?.
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