Question

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Answer 1

The statement that is true about the function is D. it is discontinuous and non-differentiable at x = 3.

How to determine which statement is true?

To determine which statement is true, we need to know the conditions for continuity and differentiablity of a function.

Conditions for continuity and differentiablity of a function.

• For a function f(x) to be continuous at a point x = a, then both the left hand limit of f(x) and the right hand limit of f(x) as x → a must be equal. That is $\lim_{x \to a^{-} } f(x) = \lim_{x \to a^{+} } f(x)$. So,  $\lim_{x \to a^{} } f(x)$ must exist since  $\lim_{x \to a^{-} } f(x) = \lim_{x \to a^{+} } f(x) = \lim_{x \to a^{} } f(x)$

• Also, for a function to be differentiable at a point x = a, it must also exist at x = a

So, since f(x) = {x² - 1 if -1 ≤ x ≤ 3 and x²/3 if 3 < x ≤ 8}

From the equality on the first condition,we see that f(x) is exists at x = 3 but is not continuous since f(x) changes to another function when x > 3. So,left hand limit of f(x) and the right hand limit of f(x) as x → 3 are not equal.

That is $\lim_{x \to 3^{-} } f(x) \neq \lim_{x \to 3^{+} } f(x)$ . Thus, the function is discontinuous at x = 3.

For differentiability, both conditions must be met. Since only one condition is met, it is non-differentiable.

So, the function is discontinuous and non-differentiable at x = 3.

So, the statement that is true about the function is D. it is discontinuous and non-differentiable at x = 3.

Learn more about continuity of a function here:

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