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Consider the function y = | x | in the interval [-1, 1], In this interval, the function is
- a)Continuous and differentiable
- b)Continuous but not differentiable
- c)Differentiable but not continuous
- d)Neither continuous nor differentiable
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the function y = | x | in the interval [-1, 1], In this inter...
The function y = | x | in the interval [-1, 1 ] is
| x | is continuous and differentiable every where except at x = 0, where it is continuous but not differentiable.
Since [-1, 1] contains 0, in this interval it is continuous but not differentiable.
| x | is continuous and differentiable every where except at x = 0, where it is continuous but not differentiable.
Since [-1, 1] contains 0, in this interval it is continuous but not differentiable.
Most Upvoted Answer
Consider the function y = | x | in the interval [-1, 1], In this inter...
Explanation:
The function y = |x| is defined as the absolute value of x. In the interval [-1, 1], this function can be represented as:
y = |x| = x for x ≥ 0
y = |x| = -x for x < />
Continuous:
A function is said to be continuous if there are no abrupt changes or jumps in its graph. In other words, the graph of a continuous function can be drawn without lifting the pen from the paper.
In the case of the function y = |x|, it is continuous for all values of x except at x = 0. At x = 0, there is a sharp point or cusp in the graph, which indicates a jump in the function. This means that the function is not continuous at x = 0.
Therefore, the function y = |x| is not continuous in the interval [-1, 1].
Differentiable:
A function is said to be differentiable if it has a derivative at every point in its domain. The derivative of a function represents its rate of change.
In the case of the function y = |x|, the derivative is not defined at x = 0. At x = 0, the function has a sharp corner or cusp, and the slope of the function changes abruptly. This means that the function is not differentiable at x = 0.
Therefore, the function y = |x| is not differentiable in the interval [-1, 1].
Conclusion:
In the interval [-1, 1], the function y = |x| is neither continuous nor differentiable. The function has a sharp point or cusp at x = 0, which makes it discontinuous, and the derivative is not defined at x = 0, which makes it non-differentiable.
Hence, the correct answer is option B) Continuous but not differentiable.
The function y = |x| is defined as the absolute value of x. In the interval [-1, 1], this function can be represented as:
y = |x| = x for x ≥ 0
y = |x| = -x for x < />
Continuous:
A function is said to be continuous if there are no abrupt changes or jumps in its graph. In other words, the graph of a continuous function can be drawn without lifting the pen from the paper.
In the case of the function y = |x|, it is continuous for all values of x except at x = 0. At x = 0, there is a sharp point or cusp in the graph, which indicates a jump in the function. This means that the function is not continuous at x = 0.
Therefore, the function y = |x| is not continuous in the interval [-1, 1].
Differentiable:
A function is said to be differentiable if it has a derivative at every point in its domain. The derivative of a function represents its rate of change.
In the case of the function y = |x|, the derivative is not defined at x = 0. At x = 0, the function has a sharp corner or cusp, and the slope of the function changes abruptly. This means that the function is not differentiable at x = 0.
Therefore, the function y = |x| is not differentiable in the interval [-1, 1].
Conclusion:
In the interval [-1, 1], the function y = |x| is neither continuous nor differentiable. The function has a sharp point or cusp at x = 0, which makes it discontinuous, and the derivative is not defined at x = 0, which makes it non-differentiable.
Hence, the correct answer is option B) Continuous but not differentiable.
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Consider the function y = | x | in the interval [-1, 1], In this interval, the function isa)Continuous and differentiableb)Continuous but not differentiablec)Differentiable but not continuousd)Neither continuous nor differentiableCorrect answer is option 'B'. Can you explain this answer?
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