The problem presented in the image states the following:

If

$\lim_{{x \to 3}} \frac{g(3) - g(x)}{3 - x} = -0.628,$

then near the point where $x = 3$, the graph of $g(x)$:

(A) is decreasing
(B) is increasing
(C) is concave downwards
(D) has a point of inflection

Solution:

The limit expression

$\lim_{{x \to 3}} \frac{g(3) - g(x)}{3 - x}$

is the definition of the derivative of $g(x)$ at $x = 3$, except with a negative sign due to the order of the terms. Normally, the derivative is written as:

$g'(3) = \lim_{{x \to 3}} \frac{g(x) - g(3)}{x - 3}.$

The expression given in the problem is essentially $-g'(3)$, so we conclude:

$-g'(3) = -0.628 \quad \text{or} \quad g'(3) = 0.628.$

Since $g'(3) = 0.628$, which is positive, the function $g(x)$ is increasing at $x = 3$.

Thus, the correct answer is:

$\boxed{B}.$

Would you like further explanation or have any other questions?

Here are 5 related questions to deepen your understanding:

1. What is the geometric interpretation of the derivative of a function at a point?
2. How does the sign of the first derivative influence the increasing or decreasing nature of a function?
3. What are the conditions for concavity, and how is the second derivative used to determine concavity?
4. Can a function be increasing and concave down at the same point?
5. How can you determine if a point is a point of inflection?

Tip:

Always remember that the first derivative provides information about the slope of the graph, while the second derivative tells you about the concavity.