Question

if y is a function of x such that $y>0$ for all x and $y<0$ for all x, which of the following could be part of the graph of $y = f(x)$? (options with graphs labeled a, b, c, d, e)

Explanation:

Step1: Analyze \( y' > 0 \)

A positive first derivative (\( y' > 0 \)) means the function \( y = f(x) \) is increasing for all \( x \). So we can eliminate graphs that are decreasing. Options C and D show decreasing functions (as \( x \) increases, \( y \) decreases), so we rule out C and D.

Step2: Analyze \( y'' < 0 \)

A negative second derivative (\( y'' < 0 \)) means the function is concave down for all \( x \). Concave down graphs curve downward (like a cup turned upside - down). Let's analyze the remaining options:

• Option A: The graph is curving upward (concave up), because the slope of the tangent lines is increasing. So this is concave up (\( y'' > 0 \)), which does not satisfy \( y'' < 0 \).
• Option B: The graph is curving downward. As \( x \) increases, the slope of the tangent lines is decreasing, which means it is concave down (\( y'' < 0 \)), and it is also increasing (\( y' > 0 \)) as \( x \) increases.
• Option E: The graph is curving upward (concave up), since the slope of the tangent lines is increasing, so \( y'' > 0 \), which does not satisfy the condition.

Answer:

B. The graph in option B