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?

पर्याय

• 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.

Answer 1

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`.