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?

Calculation:

Given,

The function is \( f(x) = x^2 + 9 \).

Statement I: f(x) is an increasing function.

The derivative of f(x) is:

\( f'(x) = \frac{d}{dx}(x^2 + 9) = 2x \)

When \(( x > 0 )\), \(( f'(x) > 0 )\), so (f(x) is increasing.

When (x < 0), f'(x) < 0 ), so f(x)  is decreasing.

At (x = 0), ( f'(x) = 0 ), meaning the function is neither increasing nor decreasing at this point.

Hence, f(x)  is not entirely increasing. It is increasing for (x > 0) and decreasing for ( x < 0).

Statement II: f(x) has local maximum at x = 0

Since the function\( f(x) = x^2 + 9 \) is a parabola opening upwards (because the coefficient of x² is positive), it has a global minimum at x = 0, not a local maximum.

Conclusion:

- Statement I is incorrect because the function is not entirely increasing. It is increasing for x > 0  and decreasing for  x < 0 .

- Statement II is incorrect because the function has a global minimum at x = 0, not a local maximum.

Hence, the correct answer is Option 4.