Question

at the point shown on the function above, which of the following is true?\
\\( \bigcirc \\ f< 0, f> 0 \\)\
\\( \bigcirc \\ f< 0, f< 0 \\)\
\\( \bigcirc \\ f> 0, f< 0 \\)\
\\( \bigcirc \\ f> 0, f> 0 \\)

Explanation:

Step1: Analyze \( f' \) (first derivative)

The first derivative \( f' \) represents the slope of the tangent line to the function. At the black dot (the point in question), we observe the direction of the function's graph. Moving from left to right, the graph is increasing (going up) at this point. An increasing function has a positive slope, so \( f' > 0 \).

Step2: Analyze \( f'' \) (second derivative)

The second derivative \( f'' \) tells us about the concavity of the function. If \( f'' > 0 \), the function is concave up (like a cup); if \( f'' < 0 \), it's concave down (like a cap). At the black dot, the graph is part of a "cup - shaped" (concave up) region? Wait, no—wait, the point is on the right - hand minimum - like part? Wait, looking at the graph, the point is on the right - hand curve. Wait, actually, the graph has a local minimum around there? Wait, no, the black dot is on the right - hand part where the function is curving upwards? Wait, no, let's re - examine. The function, at the black dot, the curve is concave up (because the second derivative for a concave up function is positive). Wait, no—wait, when the function is increasing and concave up, \( f' > 0 \) and \( f'' > 0 \)? Wait, no, wait the options: let's check again. Wait, the point is on the right - hand side, after the local minimum. Wait, the graph: from left to right, after the local minimum (the lowest point before increasing again), the function is increasing (so \( f' > 0 \)) and the concavity: since the graph is curving upwards (concave up), \( f'' > 0 \)? Wait, no, wait the options. Wait, the correct analysis:

Wait, the black dot is on the right - hand part of the graph. Let's see the slope: as we move from left to right through the black dot, the function is going up, so the slope (first derivative) is positive (\( f' > 0 \)). Now, the concavity: the graph at that point is concave up (because the curve is opening upwards, like a parabola opening upwards), so the second derivative \( f'' > 0 \). Wait, but let's check the options. The options are:

1. \( f' < 0, f'' > 0 \)
2. \( f' < 0, f'' < 0 \)
3. \( f' > 0, f'' < 0 \)
4. \( f' > 0, f'' > 0 \)

So if the function is increasing (slope positive) and concave up (second derivative positive), then the correct option is \( f' > 0, f'' > 0 \).

Wait, maybe I made a mistake earlier. Let's re - look at the graph. The black dot is on the right - hand side, after the local minimum. The function is increasing (so \( f' > 0 \)) and the graph is concave up (because the curve is curving upwards, so \( f'' > 0 \)). So the correct option is the last one.

Answer:

\( f' > 0, f'' > 0 \) (the option with \( f' > 0, f'' > 0 \))