Questions: The function f(x) is continuous on (-∞, ∞). Use the given information to sketch the graph of f. - f(0)=12, f(6)=0, f(12)=-12 ; f′(0)=0, f′(12)=0 - f′(x)>0 on (-∞, 0) and (12, ∞) ; f′(x)<0 on (0,12) - f′′(6)=0 - f′′(x)>0 on (6, ∞) ; f′′(x)<0 on (-∞, 6) Choose the correct graph of f below. A. B. C. D.

Solution Steps

We are given several pieces of information about the function f(x):

• Function values: f(0) = 12, f(6) = 0, f(12) = -12. This gives us three points on the graph: (0, 12), (6, 0), and (12, -12).

• First derivative: f'(x) > 0 on (-∞, 0) and (12, ∞), which means f(x) is increasing on these intervals. f'(x) < 0 on (0, 12), meaning f(x) is decreasing on this interval. f'(0) = 0 and f'(12) = 0, indicating that x = 0 and x = 12 are critical points, and specifically local extrema. Since the function goes from increasing to decreasing at x = 0, there is a local maximum at (0, 12). Since it goes from decreasing to increasing at x = 12, there is a local minimum at (12, -12).

• Second derivative: f''(x) < 0 on (-∞, 6), indicating that f(x) is concave down on this interval. f''(x) > 0 on (6, ∞), meaning f(x) is concave up on this interval. f''(6) = 0 means there is an inflection point at x = 6.

• Option A: This graph is increasing from roughly x = -20 up to around x = 10 and then decreases. It does not exhibit the correct increasing/decreasing behavior based on the first derivative information.

• Option B: This graph has more of a wave pattern which does not match any of the given information.

• Option D: This graph increases then decreases, and increases again but has the inflection point occurring before the local minimum.

• Option C: This graph matches all the provided information. It starts by increasing, reaches a maximum at approximately (0, 12), then decreases to a minimum around (12, -12), and increases again from there. It starts concave down and switches to concave up around x=6 where f(6)=0

Final Answer: C