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Question
Consider the graph `y = x^(1/3)`
Statement 1: The above graph is continuous at x = 0
Statement 2: The above graph is differentiable at x = 0
Which of the following is correct?
Options
Statement 1 is true and Statement 2 is false.
Statement 2 is true and Statement 1 is false.
Both the statements are true.
Both the statements are false.
Solution
Statement 1 is true and Statement 2 is false.
Explanation:
Statement 1: A function f(x) is continuous at x = 0 If:
`lim_(x rightarrow 0)f(x) = f(0)`
For `y = x^(1//3)`, we have:
`lim_(x rightarrow0)x^(1//3) = 0^(1//3) = 0`
Since `f(0) = 0^(1//3) = 0`, the limit equals the function value.
Therefore, `y = x^(1//3)` is continuous at `x = 0`.
Statement 2: A function `f(x)` is differentiable at `x = 0` if the derivative exists at that point.
The derivative of `y = x^(1//3)` is given by:
`dy/dx = d/dx(x^(1//3))`
= `1/3x^(-2//3)`
Evaluating the derivative at `x = 0`:
\[\left.\frac{1}{3}x^{-2/3}\right|_{x=0}\]
As `x rightarrow 0, x^(-2//3) rightarrow oo`.
Therefore, the derivative does not exist at `x = 0`.
Hence, `y = x^(1//3)` is not differentiable at `x = 0`.
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