Consider the following for the two (02) items that follow:
Let the function $f(x) = x^2 + 9$.

Consider the following statements:
I. $f(x)$ is an increasing function.
II. $f(x)$ has local maximum at $x = 0$.
Which of the statements given above is/are correct?

Given $f(x) = x^2 + 9$.

Differentiating with respect to $x$, we get $f'(x) = 2x$.

For $x < 0$, $f'(x) < 0$, which means $f(x)$ is strictly decreasing in $(-\infty, 0)$.

For $x > 0$, $f'(x) > 0$, which means $f(x)$ is strictly increasing in $(0, \infty)$.

Since $f(x)$ is not increasing on its entire domain, Statement I is incorrect.

To find local extrema, we set $f'(x) = 0$, which gives $x = 0$.

The second derivative is $f''(x) = 2$.

Since $f''(0) = 2 > 0$, $f(x)$ has a local minimum at $x = 0$, not a local maximum.

Thus, Statement II is also incorrect.

Both statements are incorrect.

Answer: Neither I nor II