Mechanical Engineering Exam > Mechanical Engineering Questions > The function f(x) = |x+1| on the interval [-2...
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The function f(x) = |x+1| on the interval [-2, 0]
- a)Continuous and differentiable
- b)Continuous on the integral but not differentiable at all points
- c)Neither continuous nor differentiable
- d)Differentiable but not continuous
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and diff...
f(x ) = x+ 1
f is continuous in [−2, 0]
but not differentiable at
x =−1 because we can draw
infinite number of tangents at x = −1
Most Upvoted Answer
The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and diff...
Explanation:
The function f(x) = |x 1| on the interval [-2, 0] is a piecewise function, which means it has a different formula for different parts of the interval. In this case, the function is defined as:
f(x) = -x-1 for x < />
f(x) = x+1 for x >= -1
To determine if the function is continuous and differentiable on the interval [-2, 0], we need to check the properties of continuity and differentiability at every point in the interval.
Continuity:
A function is continuous at a point x if the limit of the function at x exists and is equal to the value of the function at x. A function is continuous on an interval if it is continuous at every point in the interval.
In this case, the function is continuous on the interval [-2, 0] because the limit of the function exists and is equal to the value of the function at every point in the interval. Specifically:
lim x->-1- f(x) = lim x->-1- (-x-1) = 0
lim x->-1+ f(x) = lim x->-1+ (x+1) = 0
f(-1) = 0
Therefore, f(x) is continuous at x = -1.
At x = -2 and x = 0, the function is also continuous because it is constant (f(-2) = -1 and f(0) = 1).
Differentiability:
A function is differentiable at a point x if the limit of the difference quotient exists as x approaches the point. A function is differentiable on an interval if it is differentiable at every point in the interval.
In this case, the function is not differentiable at x = -1 because the left and right limits of the difference quotient are not equal. Specifically:
f'(x) = -1 for x < />
f'(x) = 1 for x > -1
Therefore, f(x) is not differentiable at x = -1.
At x = -2 and x = 0, the function is differentiable because it is constant (f'(-2) = 0 and f'(0) = 0).
Conclusion:
Based on the above analysis, we can conclude that the function f(x) = |x 1| on the interval [-2, 0] is continuous on the interval but not differentiable at x = -1. Therefore, the correct answer is option 'B'.
The function f(x) = |x 1| on the interval [-2, 0] is a piecewise function, which means it has a different formula for different parts of the interval. In this case, the function is defined as:
f(x) = -x-1 for x < />
f(x) = x+1 for x >= -1
To determine if the function is continuous and differentiable on the interval [-2, 0], we need to check the properties of continuity and differentiability at every point in the interval.
Continuity:
A function is continuous at a point x if the limit of the function at x exists and is equal to the value of the function at x. A function is continuous on an interval if it is continuous at every point in the interval.
In this case, the function is continuous on the interval [-2, 0] because the limit of the function exists and is equal to the value of the function at every point in the interval. Specifically:
lim x->-1- f(x) = lim x->-1- (-x-1) = 0
lim x->-1+ f(x) = lim x->-1+ (x+1) = 0
f(-1) = 0
Therefore, f(x) is continuous at x = -1.
At x = -2 and x = 0, the function is also continuous because it is constant (f(-2) = -1 and f(0) = 1).
Differentiability:
A function is differentiable at a point x if the limit of the difference quotient exists as x approaches the point. A function is differentiable on an interval if it is differentiable at every point in the interval.
In this case, the function is not differentiable at x = -1 because the left and right limits of the difference quotient are not equal. Specifically:
f'(x) = -1 for x < />
f'(x) = 1 for x > -1
Therefore, f(x) is not differentiable at x = -1.
At x = -2 and x = 0, the function is differentiable because it is constant (f'(-2) = 0 and f'(0) = 0).
Conclusion:
Based on the above analysis, we can conclude that the function f(x) = |x 1| on the interval [-2, 0] is continuous on the interval but not differentiable at x = -1. Therefore, the correct answer is option 'B'.
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The function f(x) = |x+1| on the interval [-2, 0]a)Continuous and differentiable b)Continuous on the integral but not differentiable at all points c)Neither continuous nor differentiable d)Differentiable but not continuousCorrect answer is option 'B'. Can you explain this answer?
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